10. ELECTROWEAK MODEL AND CONSTRAINTSON NEW PHYSICS · 2010. 7. 30. · 10.1 Introduction 10.2...

10. Electroweak model and constraints on new physics 1

10. ELECTROWEAK MODEL ANDCONSTRAINTS ON NEW PHYSICS

Revised November 2009 by J. Erler (U. Mexico) and P. Langacker (Institute for AdvancedStudy).

10.1 Introduction10.2 Renormalization and radiative corrections10.3 Low energy electroweak observables10.4 W and Z boson physics10.5 Precision flavor physics10.6 Experimental results10.7 Constraints on new physics

10.1. Introduction

The standard electroweak model (SM) is based on the gauge group [1] SU(2) × U(1),with gauge bosons W i

μ, i = 1, 2, 3, and Bμ for the SU(2) and U(1) fac-tors, respectively, and the corresponding gauge coupling constants g and g′.The left-handed fermion fields of the ith fermion family transform as doublets

Ψi =(

νi

�−i

)and

(uid′i

)under SU(2), where d′i ≡ ∑

j Vij dj , and V is the Cabibbo-

Kobayashi-Maskawa mixing matrix. (Constraints on V and tests of universality arediscussed in Ref. 2 and in the Section on “The CKM Quark-Mixing Matrix”. Theextension of the formalism to allow an analogous leptonic mixing matrix is discussed inthe Section on “Neutrino Mass, Mixing, and Flavor Change”.) The right-handed fieldsare SU(2) singlets. In the minimal model there are three fermion families and a single

complex Higgs doublet φ ≡(

φ+

φ0

)which is introduced for mass generation.

After spontaneous symmetry breaking the Lagrangian for the fermion fields, ψi, is

LF =∑

i

ψi

(i �∂ − mi − gmiH

2MW

)ψi

− g

2√

2

∑i

Ψi γμ (1 − γ5)(T+ W+μ + T− W−

μ ) Ψi

− e∑

i

qi ψi γμ ψi Aμ

− g

2 cos θW

∑i

ψi γμ(giV − gi

Aγ5) ψi Zμ . (10.1)

θW ≡ tan−1(g′/g) is the weak angle; e = g sin θW is the positron electric charge; andA ≡ B cos θW + W 3 sin θW is the (massless) photon field. W± ≡ (W 1 ∓ iW 2)/

√2 and

Z ≡ −B sin θW + W 3 cos θW are the massive charged and neutral weak boson fields,

K. Nakamura et al., JPG 37, 075021 (2010) (http://pdg.lbl.gov)July 30, 2010 14:36

2 10. Electroweak model and constraints on new physics

respectively. T+ and T− are the weak isospin raising and lowering operators. The vectorand axial-vector couplings are

giV ≡t3L(i) − 2qi sin2 θW , (10.2a)

giA ≡t3L(i), (10.2b)

where t3L(i) is the weak isospin of fermion i (+1/2 for ui and νi; −1/2 for di and ei) andqi is the charge of ψi in units of e.

The second term in LF represents the charged-current weak interaction [3,4]. Forexample, the coupling of a W to an electron and a neutrino is

− e

2√

2 sin θW

[W−

μ e γμ(1 − γ5)ν + W+μ ν γμ (1 − γ5)e

]. (10.3)

For momenta small compared to MW , this term gives rise to the effective four-fermioninteraction with the Fermi constant given (at tree level, i.e., lowest order in perturbationtheory) by GF /

√2 = g2/8M2

W . CP violation is incorporated in the SM by a singleobservable phase in Vij . The third term in LF describes electromagnetic interactions(QED), and the last is the weak neutral-current interaction.

In Eq. (10.1), mi is the mass of the ith fermion ψi. For the quarks these are thecurrent masses. For the light quarks, as described in the note on “Quark Masses” inthe Quark Listings, mu = 2.5+0.8

−1.0 MeV, md = 5.0+1.0−1.5 MeV, and ms = 105+25

−35 MeV.These are running MS masses evaluated at the scale μ = 2 GeV. (In this Section wedenote quantities defined in the modified minimal subtraction (MS) scheme by a caret;the exception is the strong coupling constant, αs, which will always correspond to theMS definition and where the caret will be dropped.) For the heavier quarks we useQCD sum rule constraints [5] and recalculate their masses in each call of our fits toaccount for their direct αs dependence. We find, mc(μ = mc) = 1.266+0.031

−0.036 GeV andmb(μ = mb) = 4.198 ± 0.023 GeV, with a correlation of 25%. The top quark “pole”mass, mt = 173.1± 1.3 GeV, is an average [6] of published and preliminary CDF and DØresults from run I and II. We are working, however, with MS masses in all expressions tominimize theoretical uncertainties, and therefore convert this result to the top quark MS

mass,

mt(μ = mt) = mt[1 − 43

αs

π+ O(α2

s)],

using the three-loop formula [7]. This introduces an additional uncertainty which weestimate to 0.5 GeV (the size of the three-loop term). We are assuming that the kinematicmass extracted from the collider events corresponds within this uncertainty to the polemass. Using the BLM optimized [8] version of the two-loop perturbative QCD formula [9](as we did in previous editions of this Review) gives virtually identical results. Thus,we will use mt = 173.1 ± 0.6 (stat.) ± 1.1 (syst.) ± 0.5 (QCD) GeV ≈ 173.1 ± 1.35 GeV(together with MH = 117 GeV) for the numerical values quoted in Sec. 10.2–Sec. 10.5.In the presence of right-handed neutrinos, Eq. (10.1) gives rise also to Dirac neutrinomasses. The possibility of Majorana masses is discussed in the Section on “NeutrinoMass, Mixing, and Flavor Change”.

July 30, 2010 14:36


H is the physical neutral Higgs scalar which is the only remaining part of φ afterspontaneous symmetry breaking. The Yukawa coupling of H to ψi, which is flavordiagonal in the minimal model, is gmi/2MW . In non-minimal models there are additionalcharged and neutral scalar Higgs particles [10].

10.2. Renormalization and radiative corrections

The SM has three parameters (not counting the Higgs boson mass, MH , and thefermion masses and mixings). A particularly useful set contains the Z mass, the finestructure constant, and the Fermi constant, which will be discussed in turn:

The Z boson mass, MZ = 91.1876 ± 0.0021 GeV, has been determined from theZ lineshape scan at LEP 1 [11].

The fine structure constant, α = 1/137.035999084(51), is currently best determinedfrom the e± anomalous magnetic moment [12] using the revised 4-loop QED result fromRef. 13. (For other determinations, see Ref. 14.) In most electroweak renormalizationschemes, it is convenient to define a running α dependent on the energy scale of theprocess, with α−1 ∼ 137 appropriate at very low energy, i.e. close to the Thomsonlimit. (The running has also been observed [15] directly.) For scales above a few hundredMeV this introduces an uncertainty due to the low energy hadronic contribution tovacuum polarization. In the modified minimal subtraction (MS) scheme [16] (usedfor this Review), and with αs(MZ) = 0.120 for the QCD coupling at MZ , we haveα(mτ )−1 = 133.444± 0.015 and α(MZ)−1 = 127.916± 0.015. The latter corresponds to aquark sector contribution (without the top) to the conventional (on-shell) QED coupling,α(MZ) =

α

1 − Δα(MZ), of Δα

(5)had(MZ) ≈ 0.02793 ± 0.00011. These values are updated

from Ref. 17 with Δα(5)had(MZ) moved upwards and its uncertainty almost halved (mostly

due to a more precise mc(mc)). Its correlation with the μ± anomalous magnetic moment(see Sec. 10.5), as well as the non-linear αs dependence of α(MZ) and the resultingcorrelation with the input variable αs, are fully taken into account in the fits. This isdone by using as actual input (fit constraint) instead of Δα

(5)had(MZ) the analogous low

energy contribution by the three light quarks, Δα(3)had(1.8 GeV) = (57.29 ± 0.90) × 10−4,

and by calculating the perturbative and heavy quark contributions to α(MZ) in each callof the fits according to Ref. 17. The uncertainty is from e+e− annihilation data below1.8 GeV and τ decay data; from isospin breaking effects (affecting the interpretationof the τ data); from uncalculated higher order perturbative and non-perturbative QCDcorrections; and from the MS quark masses. Such a short distance mass definition (unlikethe pole mass) is free from non-perturbative and renormalon [18] uncertainties. Variousrecent evaluations of Δα

(5)had are summarized in Table 10.1. where the leading order

relation between the MS and on-shell definitions is given by,

Δα(MZ) − Δα(MZ) =α

π

(10027

− 16− 7

4ln

M2Z

M2W

)≈ 0.0072,

July 30, 2010 14:36


Table 10.1: Recent evaluations of the on-shell Δα(5)had(MZ). For better comparison

we adjusted central values and errors to correspond to a common and fixed value ofαs(MZ) = 0.120. References quoting results without the top quark decoupled areconverted to the five flavor definition. Ref. [29] uses ΛQCD = 380± 60 MeV; for theconversion we assumed αs(MZ) = 0.118 ± 0.003.

Reference Result Comment

Martin, Zeppenfeld [19] 0.02744 ± 0.00036 PQCD for√

s > 3 GeV

Eidelman, Jegerlehner [20] 0.02803 ± 0.00065 PQCD for√

s > 40 GeV

Geshkenbein, Morgunov [21] 0.02780 ± 0.00006 O(αs) resonance model

Burkhardt, Pietrzyk [22] 0.0280 ± 0.0007 PQCD for√

s > 40 GeV

Swartz [23] 0.02754 ± 0.00046 use of fitting function

Alemany et al. [24] 0.02816 ± 0.00062 incl. τ decay data

Krasnikov, Rodenberg [25] 0.02737 ± 0.00039 PQCD for√

s > 2.3 GeV

Davier & Hocker [26] 0.02784 ± 0.00022 PQCD for√

s > 1.8 GeV

Kuhn & Steinhauser [27] 0.02778 ± 0.00016 complete O(α2s)

Erler [17] 0.02779 ± 0.00020 conv. from MS scheme

Davier & Hocker [28] 0.02770 ± 0.00015 use of QCD sum rules

Groote et al. [29] 0.02787 ± 0.00032 use of QCD sum rules

Martin et al. [30] 0.02741 ± 0.00019 incl. new BES data

Burkhardt, Pietrzyk [31] 0.02763 ± 0.00036 PQCD for√

s > 12 GeV

de Troconiz, Yndurain [32] 0.02754 ± 0.00010 PQCD for s > 2 GeV2

Jegerlehner [33] 0.02765 ± 0.00013 conv. from MOM scheme

Hagiwara et al. [34] 0.02757 ± 0.00023 PQCD for√

s > 11.09 GeV

Burkhardt, Pietrzyk [35] 0.02760 ± 0.00035 incl. KLOE data

Hagiwara et al. [36] 0.02770 ± 0.00022 incl. selected KLOE data

Jegerlehner [37] 0.02755 ± 0.00013 Adler function approach

and where the first term is from fermions and the other two are from W± loops which areusually excluded from the on-shell definition. Most of the older results relied on e+e− →hadrons cross-section measurements up to energies of 40 GeV, which were somewhathigher than the QCD prediction, suggested stronger running, and were less precise. Themost recent results typically assume the validity of perturbative QCD (PQCD) at scalesof 1.8 GeV and above, and are in reasonable agreement with each other. (Evaluationsin the on-shell scheme utilize threshold data from BES [38] as further input.) There is,

July 30, 2010 14:36


however, some discrepancy between analyzes based on e+e− → hadrons cross-sectiondata and those based on τ decay spectral functions [39,40]. The latter utilize data fromOPAL [41], CLEO [42], ALEPH [43], and Belle [44] and imply lower central values forthe extracted MH of about 6%. This discrepancy is smaller than in the past and at leastsome of it appears to be experimental. The dominant e+e− → π+π− cross-section wasmeasured with the CMD-2 [45] and SND [46] detectors at the VEPP-2M e+e− colliderat Novosibirsk and the results are (after an initial discrepancy due to a flaw in theMonte Carlo event generator used by SND) in good agreement with each other. As analternative to cross-section scans, one can use the high statistics radiative return events ate+e− accelerators operating at resonances such as the Φ or the Υ(4S). The method [47]is systematics dominated. The BaBar collaboration [48] studied multi-hadron eventsradiatively returned from the Υ(4S), reconstructing the radiated photon and normalizingto μ±γ final states. Their result is higher compared to VEPP-2M and in fact agreesquite well with the τ analysis including the energy dependence (shape). In contrast, theshape and smaller overall cross-section from the π+π− radiative return results from theΦ obtained by the KLOE collaboration [49] differs significantly from what is observed byBaBar. The discrepancy originates from the kinematic region

√s� 0.6 GeV, and is most

pronounced for√

s � 0.85 GeV. For a recent review on these e+e− data, see Ref. 50. Allmeasurements including older data [51] are accounted for in the fits on the basis of resultsin Refs. [28,40,50]. Further improvement of this dominant theoretical uncertainty in theinterpretation of precision data will require better measurements of the cross-section fore+e− → hadrons below the charmonium resonances including multi-pion and other finalstates. To improve the precisions in mc(mc) and mb(mb) it would help to remeasure thethreshold regions of the heavy quarks as well as the electronic decay widths of the narrowcc and bb resonances.

The Fermi constant, GF = 1.166364(5) × 10−5 GeV−2, is derived from the muonlifetime formula∗,

τ−1μ =

G2F m5

μ

192π3F (ρ)

(1 +

35

m2μ

M2W

)[1 + H1(ρ)

α(mμ)π

+ H2(ρ)α2(mμ)

π2

], (10.4)

where ρ = m2e/m2

μ, and where

F (ρ) = 1 − 8ρ + 8ρ3 − ρ4 − 12ρ2 ln ρ = 0.999813,

H1(ρ) =258

− π2

2−

(9 + 4π2 + 12 lnρ

)ρ

+ 16π2ρ3/2 + O(ρ2) = −1.8079,

∗ In the spirit of the Fermi theory, the propagator correction proportional to m2μ could

instead be incorporated into Δr (see below), but we choose to leave it in the definition ofGF for historical consistency.

July 30, 2010 14:36


H2(ρ) =1568155184

− 51881

π2 − 89536

ζ(3) +67720

π4

+536

π2 ln 2 − 54π2√ρ + O(ρ) = 6.7,

α(mμ)−1 = α−1 +13π

ln ρ = 135.9 .

The massless corrections to H1 and H2 have been obtained in Refs. 52 and 53,respectively. The mass corrections to H1 have been known for some time [54], whilethose to H2 are very recent [55]. Notice the term linear in me whose appearance wasunforeseen and can be traced to the use of the muon pole mass in the prefactor [55].The remaining uncertainty in GF is experimental and has recently been halved by theMuLan [56] and FAST [57] collaborations.

With these inputs, sin2 θW and the W boson mass, MW , can be calculated when valuesfor mt and MH are given; conversely (as is done at present), MH can be constrainedby sin2 θW and MW . The value of sin2 θW is extracted from Z pole observables andneutral-current processes [11,60], and depends on the renormalization prescription.There are a number of popular schemes [61–68] leading to values which differ by smallfactors depending on mt and MH . The notation for these schemes is shown in Table 10.2.

Table 10.2: Notations used to indicate the various schemes discussed in the text.Each definition of sin2 θW leads to values that differ by small factors depending onmt and MH . Approximate values are also given for illustration.

Scheme Notation Value

On-shell s2W 0.2233

NOV s2MZ

0.2311

MS s2Z 0.2313

MS ND s2ND 0.2315

Effective angle s2f 0.2316

(i) The on-shell scheme [61] promotes the tree-level formula sin2 θW = 1 − M2W /M2

Z toa definition of the renormalized sin2 θW to all orders in perturbation theory, i.e.,sin2 θW → s2

W ≡ 1 − M2W /M2

Z :

MW =A0

sW (1 − Δr)1/2, MZ =

MW

cW, (10.5)

where cW ≡ cos θW , A0 = (πα/√

2GF )1/2 = 37.28061(8) GeV, and Δr includesthe radiative corrections relating α, α(MZ), GF , MW , and MZ . One finds

July 30, 2010 14:36


Δr ∼ Δr0 − ρt/ tan2 θW , where Δr0 = 1 − α/α(MZ) = 0.06655(11) is due to therunning of α, and ρt = 3GF m2

t /8√

2π2 = 0.00939(mt/173.1 GeV)2 represents thedominant (quadratic) mt dependence. There are additional contributions to Δrfrom bosonic loops, including those which depend logarithmically on MH . One hasΔr = 0.0362 ∓ 0.0005 ± 0.00011, where the second uncertainty is from α(MZ). Thusthe value of s2

W extracted from MZ includes an uncertainty (∓0.00019) from thecurrently allowed range of mt. This scheme is simple conceptually. However, therelatively large (∼ 3%) correction from ρt causes large spurious contributions inhigher orders.

(ii) A more precisely determined quantity s2MZ

[62] can be obtained from MZ byremoving the (mt, MH) dependent term from Δr [63], i.e.,

s2MZ

(1 − s2MZ

) ≡ πα(MZ)√2GF M2

Z

. (10.6)

Using α(MZ)−1 = 128.91 ± 0.02 yields s2MZ

= 0.23108 ∓ 0.00005. The small

uncertainty in s2MZ

compared to other schemes is because the mt dependence hasbeen removed by definition. However, the mt uncertainty reemerges when otherquantities (e.g., MW or other Z pole observables) are predicted in terms of MZ .

Both s2W and s2

MZdepend not only on the gauge couplings but also on the spontaneous-

symmetry breaking, and both definitions are awkward in the presence of any extension ofthe SM which perturbs the value of MZ (or MW ). Other definitions are motivated by thetree-level coupling constant definition θW = tan−1(g′/g):(iii) In particular, the modified minimal subtraction (MS) scheme introduces the quantity

sin2 θW (μ) ≡ g ′2(μ)/[g 2(μ) + g ′2(μ)

], where the couplings g and g′ are defined by

modified minimal subtraction and the scale μ is conveniently chosen to be MZ formany electroweak processes. The value of s 2

Z = sin2 θW (MZ) extracted from MZ isless sensitive than s2

W to mt (by a factor of tan2 θW ), and is less sensitive to mosttypes of new physics than s2

W or s2MZ

. It is also very useful for comparing withthe predictions of grand unification. There are actually several variant definitionsof sin2 θW (MZ), differing according to whether or how finite α ln(mt/MZ) termsare decoupled (subtracted from the couplings). One cannot entirely decouple theα ln(mt/MZ) terms from all electroweak quantities because mt mb breaks SU(2)symmetry. The scheme that will be adopted here decouples the α ln(mt/MZ) termsfrom the γ–Z mixing [16,64], essentially eliminating any ln(mt/MZ) dependence inthe formulae for asymmetries at the Z pole when written in terms of s 2

Z . (A similardefinition is used for α.) The various definitions are related by

s 2Z = c (mt, MH)s2

W = c (mt, MH) s2MZ

, (10.7)

where c = 1.0361 ± 0.0005 and c = 1.0010 ∓ 0.0002. The quadratic mt dependenceis given by c ∼ 1 + ρt/ tan2 θW and c ∼ 1 − ρt/(1 − tan2 θW ), respectively. Theexpressions for MW and MZ in the MS scheme are

MW =A0

sZ(1 − ΔrW )1/2, MZ =

MW

ρ 1/2 cZ

, (10.8)

July 30, 2010 14:36


and one predicts ΔrW = 0.06971 ± 0.00002 ± 0.00011. ΔrW has no quadratic mt

dependence, because shifts in MW are absorbed into the observed GF , so that theerror in ΔrW is dominated by Δr0 = 1− α/α(MZ) which induces the second quoteduncertainty. The quadratic mt dependence has been shifted into ρ ∼ 1 + ρt, whereincluding bosonic loops, ρ = 1.01047 ± 0.00015. Quadratic MH effects are deferredto two-loop order, while the leading logarithmic MH effect is a good approximationonly for large MH values which are currently disfavored by the precision data. As anillustration, the shift in MW due to a large MH (for fixed MZ) is given by

ΔHMW = −1196

α

π

MW

c2W − s2W

lnM2

H

M2W

+ O(α2)

∼ −200 MeV (for MH = 10 MW ).

(iv) A variant MS quantity s 2ND (used in the 1992 edition of this Review) does not

decouple the α ln(mt/MZ) terms [65]. It is related to s 2Z by

s 2Z = s 2

ND/(1 +

α

πd), (10.9a)

d =13

(1s 2

− 83

) [(1 +

αs

π) ln

mt

MZ− 15αs

8π

], (10.9b)

Thus, s 2Z − s 2

ND ∼ −0.0002 for mt = 173.1 GeV.

(v) Yet another definition, the effective angle [66–68] s2f for the Z vector coupling to

fermion f , is described in Sec. 10.3.Experiments are at such level of precision that complete O(α) radiative corrections

must be applied. For neutral-current and Z pole processes, these corrections areconveniently divided into two classes:

1. QED diagrams involving the emission of real photons or the exchange of virtualphotons in loops, but not including vacuum polarization diagrams. These graphsoften yield finite and gauge-invariant contributions to observable processes. However,they are dependent on energies, experimental cuts, etc., and must be calculatedindividually for each experiment.

2. Electroweak corrections, including γγ, γZ, ZZ, and WW vacuum polarizationdiagrams, as well as vertex corrections, box graphs, etc., involving virtual Wand Z bosons. Many of these corrections are absorbed into the renormalizedFermi constant defined in Eq. (10.4). Others modify the tree-level expressions forZ pole observables and neutral-current amplitudes in several ways [58]. One-loopcorrections are included for all processes. In addition, certain two-loop correctionsare also important. In particular, two-loop corrections involving the top quarkmodify ρt in ρ, Δr, and elsewhere by

ρt → ρt[1 + R(MH , mt)ρt/3]. (10.10)

July 30, 2010 14:36


R(MH , mt) is best described as an expansion in M2Z/m2

t . The unsuppressed termswere first obtained in Ref. 69, and are known analytically [70]. Contributionssuppressed by M2

Z/m2t were first studied in Ref. 71 with the help of small and large

Higgs mass expansions, which can be interpolated. These contributions are aboutas large as the leading ones in Refs. 69 and 70. The complete two-loop calculationof Δr (without further approximation) has been performed in Refs. 72 and 73for fermionic and purely bosonic diagrams, respectively. Similarly, the electroweaktwo-loop calculation for the relation between s2

� and s2W is complete [74] including

the recently obtained purely bosonic contribution [75]. For MH above its lowerdirect limit, −17 < R ≤ −13.

Mixed QCD-electroweak contributions to gauge boson self-energies of orderααsm

2t [76] and αα2

sm2t [77] increase the predicted value of mt by 6%. This is,

however, almost entirely an artifact of using the pole mass definition for mt. Theequivalent corrections when using the MS definition mt(mt) increase mt by less than0.5%. The subleading ααs corrections [78] are also included. Further three-loopcorrections of order αα2

s [79], α3m6t [80,81], and α2αsm

4t (for MH = 0) [80], are

rather small. The same is true for α3M4H [82] corrections unless MH approaches

1 TeV. Also known are the singlet contributions (pure gluonic intermediate states) oforder αα2

s [83] and αα3s [84]. Recently, the corresponding non-singlet contributions

have been computed as well [85].

The leading electroweak two-loop terms for the Z → bb-vertex of O(α2m4t ) have

been obtained in Refs. 69 and 70, and the mixed QCD-electroweak contributionsin Refs. 86 and 87. Very recently, the authors of Ref. 88 completed the two-loopelectroweak fermionic corrections to s2

b . The O(ααs)-vertex corrections involvingmassless quarks [89] add coherently, resulting in a sizable effect and shift αs(MZ)when extracted from Z lineshape observables (see Sec. 10.3) by ≈ +0.0007.

Throughout this Review we utilize electroweak radiative corrections from the programGAPP [90], which works entirely in the MS scheme, and which is independent of thepackage ZFITTER [68].

10.3. Low energy electroweak observables

It is convenient to write the four-fermion interactions relevant to ν-hadron, ν-e, as wellas parity violating e-hadron and e-e neutral-current processes in a form that is valid inan arbitrary gauge theory (assuming massless left-handed neutrinos). One has,

−L νh =GF√

2ν γμ(1 − γ5)ν

∑i

[εL(i)qi γμ(1 − γ5)qi

+ εR(i)qi γμ(1 + γ5)qi

], (10.11)

−L νe =GF√

2νμγμ(1 − γ5)νμ e γμ(gνe

V − gνeA γ5)e, (10.12)

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−L eh = −GF√2

∑i

[C1i e γμγ5e qi γμqi + C2i e γμe qi γμγ5qi

], (10.13)

−L ee = −GF√2

C2e e γμγ5e e γμe, (10.14)

where one must include the charged-current contribution for νe-e and νe-e and theparity-conserving QED contribution for electron scattering.

The SM expressions for εL,R(i), gνeV,A, and Cij are given in Table 10.3. Note, that gνe

V,Aand the other quantities are coefficients of effective four-Fermi operators, which differfrom the quantities defined in Eq. (10.2) in the radiative corrections and in the presenceof possible physics beyond the SM.

10.3.1. Neutrino scattering :A precise determination of the on-shell s2

W , which depends only very weakly on mt andMH , is obtained from deep inelastic scattering (DIS) of neutrinos from (approximately)isoscalar targets [91]. The ratio Rν ≡ σNC

νN /σCCνN of neutral-to-charged-current cross-

sections has been measured to 1% accuracy by the CDHS [92] and CHARM [93]collaborations at CERN. The CCFR [94] collaboration at Fermilab has obtained aneven more precise result, so it is important to obtain theoretical expressions for Rν andRν ≡ σNC

νN /σCCνN to comparable accuracy. Fortunately, many of the uncertainties from the

strong interactions and neutrino spectra cancel in the ratio. A large theoretical uncertaintyis associated with the c-threshold, which mainly affects σCC . Using the slow rescalingprescription [95] the central value of sin2 θW from CCFR varies as 0.0111(mc [GeV]−1.31),where mc is the effective mass which is numerically close to the MS mass mc(mc), buttheir exact relation is unknown at higher orders. For mc = 1.31 ± 0.24 GeV (determinedfrom ν-induced dimuon production [96]) this contributes ±0.003 to the total uncertaintyΔ sin2 θW ∼ ±0.004. (The experimental uncertainty is also ±0.003.) This uncertaintylargely cancels, however, in the Paschos-Wolfenstein ratio [97],

R− =σNC

νN − σNCνN

σCCνN − σCC

νN

. (10.15)

It was measured by Fermilab’s NuTeV collaboration [98] for the first time, and required ahigh-intensity and high-energy anti-neutrino beam.

A simple zeroth-order approximation is

Rν = g2L + g2

Rr, Rν = g2L +

g2R

r, R− = g2

L − g2R, (10.16)

where

g2L ≡ εL(u)2 + εL(d)2 ≈ 1

2− sin2 θW +

59

sin4 θW , (10.17a)

g2R ≡ εR(u)2 + εR(d)2 ≈ 5

9sin4 θW , (10.17b)

July 30, 2010 14:36


Table 10.3: Standard Model expressions for the neutral-current parameters forν-hadron, ν-e, and e−-scattering processes. At tree level, ρ = κ = 1, λ = 0. Ifradiative corrections are included, ρνN = 1.0081, κνN (〈Q2〉 = −20 GeV2) = 0.9972,κνN (〈Q2〉 = −35 GeV2) = 0.9964, λuL = −0.0031, λdL = −0.0025 and λR =3.7 × 10−5. For ν-e scattering, ρνe = 1.0127 and κνe = 0.9965 (at 〈Q2〉 = 0.). Foratomic parity violation and the polarized DIS experiment at SLAC, ρ′e = 0.9877,ρe = 1.0006, κ′e = 1.0026, κe = 1.0299, λ′ = −1.8 × 10−5, λu = −0.0118 andλd = 0.0029. And for polarized Møller scattering with SLAC (JLab) kinematics,λe = −0.0002 (λe = −0.0004). The dominant mt dependence is given by ρ ∼ 1 + ρt,while κ ∼ 1 (MS) or κ ∼ 1 + ρt/ tan2 θW (on-shell).

Quantity Standard Model Expression

εL(u) ρνN

(12− 2

3κνN s2

Z

)+ λuL

εL(d) ρνN

(− 1

2+ 1

3κνN s2

Z

)+ λdL

εR(u) ρνN

(− 2

3κνN s2

Z

)+ λR

εR(d) ρνN

(13

κνN s2Z

)+ 2 λR

gνeV ρνe

(− 1

2+ 2 κνe s2

Z

)gνeA ρνe

(− 1

2

)C1u ρ′e

(− 1

2+ 4

3κ′e s2

Z

)+ λ′

C1d ρ′e(

12− 2

3κ′e s2

Z

)− 2 λ′

C2u ρe

(− 1

2+ 2 κe s2

Z

)+ λu

C2d ρe

(12− 2 κe s2

Z

)+ λd

C2e ρe

(12− 2 κe s2

Z

)+ λe

and r ≡ σCCνN /σCC

νN is the ratio of ν to ν charged-current cross-sections, which canbe measured directly. (In the simple parton model, ignoring hadron energy cuts,r ≈ ( 1

3+ ε)/(1 + 1

3ε), where ε ∼ 0.125 is the ratio of the fraction of the nucleon’s

momentum carried by anti-quarks to that carried by quarks.) In practice, Eq. (10.16)must be corrected for quark mixing, quark sea effects, c-quark threshold effects,non-isoscalarity, W–Z propagator differences, the finite muon mass, QED and electroweakradiative corrections. Details of the neutrino spectra, experimental cuts, x and Q2

dependence of structure functions, and longitudinal structure functions enter only at the

July 30, 2010 14:36


level of these corrections and therefore lead to very small uncertainties. The CCFR groupquotes s2

W = 0.2236 ± 0.0041 for (mt, MH) = (175, 150) GeV with very little sensitivityto (mt, MH).

The NuTeV collaboration found s2W = 0.2277 ± 0.0016 (for the same reference values),

which was 3.0 σ higher than the SM prediction [98]. Since then a number of experimentaland theoretical developments implied shifts in the extracted s2

W , most of them reducingthe discrepancy: (i) NuTeV also measured [99] the difference between the strange andantistrange quark momentum distributions, S− ≡ ∫ 1

0 dxx[s(x)− s(x)] = 0.00196±0.00143,from dimuon events utilizing the first complete next-to-leading order QCD description [100]and parton distribution functions (PDFs) according to Ref. 101. The magnitude of thecentral value agrees with the earlier result [102] but differs in sign. The effect of S− �= 0on the NuTeV value for s2

W has been studied in Ref. 102, and the S− above translatesinto a shift δs2

W = −0.0014 ± 0.0010. On the other hand, there are theoretical arguments(see Ref. 103 and references therein) favoring a zero crossing of x[s(x) − s(x)] at valuesmuch larger than seen by NuTeV and suggesting an effect of much smaller and perhapsnegligible size. We will therefore take half of the above shift as an estimate of both theS− effect and the associated uncertainty, δs2

W = −0.0007 ± 0.0007, replacing the originaluncertainty [98] of ±0.00047. (ii) The measured branching ratio for Ke3 decays enterscrucially in the determination of the νe(νe) contamination of the νμ(νμ) beam. Thisbranching ratio has moved from 4.82 ± 0.06% at the time of the original publication [98]to the current value of 5.07 ± 0.04%, i.e., a change by more than 4 σ. To reflectthis, we move s2

W by +0.0016 and reduce the νe(νe) uncertainty by a factor of 2/3.(iii) PDFs seem to violate isospin symmetry at levels much stronger than generallyexpected [104]. A minimum χ2 set of PDFs generalized in this sense [105,106] shows areduction in the NuTeV discrepancy in s2

W by 0.0015. But isospin symmetry violatingPDFs are currently not well constrained phenomenologically and within uncertaintiesthe NuTeV anomaly could be accounted for in full or conversely made larger [105].Still, the leading contribution from quark mass differences turns out to be largelymodel-independent [107] and a shift, δs2

W = −0.0015±0.0003 [103], is applied. (iv) QEDsplitting effects also violate isospin symmetry with an effect on s2

W whose sign (reducingthe discrepancy) is model-independent. The corresponding shift of δs2

W = −0.0011 hasbeen calculated in Ref. 108 and a 100% uncertainty is assigned to it. (v) Nuclearshadowing effects [109] are likely to affect the interpretation of the NuTeV result atsome level, but the NuTeV collaboration argues that their data are dominated by valuesof Q2 at which nuclear shadowing is expected to be relatively small so that we do notapply a correction. However, another nuclear effect, the isovector EMC effect [110], ismuch larger (because it affects all neutrons in the nucleus, not just the excess ones)and model-independently works to reduce the discrepancy. It is estimated to lead toa shift of δs2

W = −0.0019 ± 0.0006 [103]. (vi) The extracted s2W may also shift at

the level of the quoted uncertainty when analyzed using the most recent QED andelectroweak radiative corrections [111,112], as well as QCD corrections to the structurefunctions [113]. However, their precise impact can be estimated only after the NuTeVdata have been analyzed with a new set of PDFs including these new radiative correctionswhile simultaneously allowing isospin breaking and asymmetric strange seas. Remaining

July 30, 2010 14:36


one- and two-loop radiative corrections have been estimated [112] to induce uncertaintiesin the extracted s2

W of ±0.0004 and ±0.0003, respectively, compared to the initial errorof ±0.00011 [102]. Most of the s2

W dependence and the NuTeV discrepancy reside ing2L (initially 2.7 σ low). Thus, the total shift of δs2

W = −0.0036 ± 0.0015 correspondsto δg2

L = +0.0027 ∓ 0.0011 and we arrive at g2L = 0.3027 ± 0.0018 which we use as a

constraint in the fits. The right-handed coupling, g2R = 0.0308 ± 0.0011 (which is 0.7 σ

high) and the other ν-DIS data are expected to exhibit shifts as well, but these oughtto be less significant since their relative experimental uncertainties are larger. In view ofthese developments and caveats, we use the NuTeV result in this Review but consider italong with the other ν-DIS data as preliminary until a re-analysis using PDFs includingall experimental and theoretical information and correlations has been completed.

The cross-section in the laboratory system for νμe → νμe or νμe → νμe elasticscattering is

dσν,ν

dy=

G2F meEν

2π

[(gνe

V ± gνeA )2 + (gνe

V ∓ gνeA )2(1 − y)2 − (gνe2

V − gνe2A )

y me

Eν

], (10.18)

where the upper (lower) sign refers to νμ(νμ), and y ≡ Te/Eν (which runs from 0 to(1 + me/2Eν)−1) is the ratio of the kinetic energy of the recoil electron to the incident νor ν energy. For Eν me this yields a total cross-section

σ =G2

F meEν

2π

[(gνe

V ± gνeA )2 +

13(gνe

V ∓ gνeA )2

]. (10.19)

The most accurate measurements [114–117] of sin2 θW from ν-lepton scattering arefrom the ratio R ≡ σνµe/σνµe in which many of the systematic uncertainties cancel.Radiative corrections (other than mt effects) are small compared to the precision ofpresent experiments and have negligible effect on the extracted sin2 θW . The mostprecise experiment (CHARM II) [116] determined not only sin2 θW but gνe

V,A as well. Thecross-sections for νe-e and νe-e may be obtained from Eq. (10.18) by replacing gνe

V,A bygνeV,A + 1, where the 1 is due to the charged-current contribution [117,118].

10.3.2. Parity violation :

The SLAC polarized electron-deuteron DIS experiment [119] measured the right-leftasymmetry,

A =σR − σL

σR + σL, (10.20)

where σR,L is the cross-section for the deep-inelastic scattering of a right- or left-handedelectron: eR,LN → eX. In the quark parton model,

A

Q2= a1 + a2

1 − (1 − y)2

1 + (1 − y)2, (10.21)

July 30, 2010 14:36


where Q2 > 0 is the momentum transfer and y is the fractional energy transfer from theelectron to the hadrons. For the deuteron or other isoscalar targets, one has, neglectingthe s-quark and anti-quarks,

a1 =3GF

5√

2πα

(C1u − 1

2C1d

)≈ 3GF

5√

2πα

(−3

4+

53

sin2 θW

), (10.22a)

a2 =3GF

5√

2πα

(C2u − 1

2C2d

)≈ 9GF

5√

2πα

(sin2 θW − 1

4

). (10.22b)

In another polarized-electron scattering experiment on deuterons, but in the quasi-elastickinematic regime, the SAMPLE experiment [120] at MIT-Bates extracted the combinationC2u − C2d at Q2 values of 0.1 GeV2 and 0.038 GeV2. What was actually determinedwere nucleon form factors from which the quoted results were obtained by the removalof a multi-quark radiative correction [121]. Other linear combinations of the Ciq havebeen determined in polarized-lepton scattering at CERN in μ-C DIS, at Mainz in e-Be(quasi-elastic), and at Bates in e-C (elastic). See the review articles in Refs. 59 and 122for more details. Recent polarized electron asymmetry experiments, i.e., SAMPLE, thePVA4 experiment at Mainz, and the HAPPEX and G0 experiments at Jefferson Lab, havefocussed on the strange quark content of the nucleon. These are reviewed in Ref. 123,where it is shown that they can also provide significant constraints on C1u and C1d whichcomplement those from atomic parity violation.

The parity violating asymmetry, APV , in fixed target polarized Møller scattering,e−e− → e−e−, is defined as in Eq. (10.20) and reads [124],

APV

Q2= −2 C2e

GF√2πα

1 − y

1 + y4 + (1 − y)4, (10.23)

where y is again the energy transfer. It has been measured at low Q2 = 0.026 GeV2

in the SLAC E158 experiment [125], with the result APV = (−1.31 ± 0.14 (stat.) ±0.10 (syst.)) × 10−7. Expressed in terms of the weak mixing angle in the MS scheme, thisyields s 2(Q2) = 0.2403± 0.0013, and established the scale dependence of the weak mixing(see Fig. 10.1) at the level of 6.4 standard deviations. One can also define the so-calledweak charge of the electron (cf. Eq. (10.24) below) as QW (e) ≡ −2 C2e = −0.0403±0.0053(the implications are discussed in Ref. 126). In a similar experiment and at about thesame Q2, Qweak at Jefferson Lab [127] will be able to measure the weak charge of theproton, QW (p) = −2 [2 C1u + C1d], and sin2 θW in polarized ep scattering with relativeprecisions of 4% and 0.3%, respectively. These experiments will provide the most precisedeterminations of the weak mixing angle off the Z peak and will be sensitive to varioustypes of physics beyond the SM.

There are precise experiments measuring atomic parity violation (APV) [131] incesium [132,133] (at the 0.4% level [132]) , thallium [134], lead [135], and bismuth [136].The electroweak physics is contained in the weak charges which are defined by,

QW (Z, N) ≡ −2 [C1u (2Z + N) + C1d(Z + 2N)] ≈ Z(1 − 4 sin2 θW ) − N. (10.24)

July 30, 2010 14:36


0.0001 0.001 0.01 0.1 1 10 100 1000 10000

μ [GeV]

0.228

0.23

0.232

0.234

0.236

0.238

0.24

0.242

0.244

0.246

0.248

0.25

sin2 θ W

(μ)

QW(APV)QW(e)

ν-DIS

LEP 1

SLC

Tevatron

e-DIS

MOLLER

Qweak

Figure 10.1: Scale dependence of the weak mixing angle defined in the MS

scheme [129] (for the scale dependence of the weak mixing angle defined in amass-dependent renormalization scheme, see Ref. 126). The minimum of the curvecorresponds to Q = MW , below which we switch to an effective theory withthe W± bosons integrated out, and where the β-function for the weak mixingangle changes sign. At the location of the W boson mass and each fermion mass,there are also discontinuities arising from scheme dependent matching terms whichare necessary to ensure that the various effective field theories within a givenloop order describe the same physics. However, in the MS scheme these are verysmall numerically and barely visible in the figure provided one decouples quarksat Q = mq(mq). The width of the curve reflects the theory uncertainty fromstrong interaction effects which at low energies is at the level of ±7 × 10−5 [129].Following the estimate [130] of the typical momentum transfer for parity violationexperiments in Cs, the location of the APV data point is given by μ = 2.4 MeV.For ν-DIS we chose μ = 20 GeV which is about half-way between the averages of√

Q2 for ν and ν interactions at NuTeV. The Tevatron measurements are stronglydominated by invariant masses of the final state dilepton pair of O(MZ) and canthus be considered as additional Z pole data points, yielding s2

Z = 0.2316 ± 0.0018.However, for clarity we displayed the point horizontally to the right.

E.g., QW (133Cs) is extracted by measuring experimentally the ratio of the parity violatingamplitude, EPNC, to the Stark vector transition polarizability, β, and by calculatingtheoretically EPNC in terms of QW . One can then write,

QW = N

(ImEPNC

β

)exp.

( |e| aB

Im EPNC

QW

N

)th.

(β

a3B

)exp.+th.

(a2B

|e|

).

July 30, 2010 14:36


The uncertainties associated with atomic wave functions are quite small for cesium [137].In the past, the semi-empirical value of β added another source of theoreticaluncertainty [138]. The ratio of the off-diagonal hyperfine amplitude to the polarizabilityhas now been measured directly by the Boulder group [139]. Combined with theprecisely known hyperfine amplitude [140] one finds, β = 26.991 ± 0.046, in excellentagreement with the earlier results, reducing the overall theory uncertainty (whileslightly increasing the experimental error). The very recent state-of-the-art many bodycalculation [141] yields, ImEPNC = (0.8906 ± 0.0026) × 10−11|e| aB QW /N , while thetwo measurements [132,133] combine to give ImEPNC/β = −1.5924 ± 0.0055 mV/cm,and we obtain QW (13378Cs) = −73.20 ± 0.35. Thus, the various theoretical efforts inRefs. 141 and 142 together with an update of the SM calculation [128] removed an earlier2.3 σ deviation from the SM (see the year 2000 edition of this Review). The theoreticaluncertainties are 3% for thallium [143] but larger for the other atoms. The Boulderexperiment in cesium also observed the parity-violating weak corrections to the nuclearelectromagnetic vertex (the anapole moment [144]) .

In the future it could be possible to further reduce the theoretical wave functionuncertainties by taking the ratios of parity violation in different isotopes [131,145].There would still be some residual uncertainties from differences in the neutron chargeradii, however [146]. Experiments in hydrogen and deterium are another possibility forreducing the uncertainties [147].

10.4. W and Z boson physics

10.4.1. e+e− scattering below the Z pole :The forward-backward asymmetry for e+e− → �+�−, � = μ or τ , is defined as

AFB ≡ σF − σB

σF + σB, (10.25)

where σF (σB) is the cross-section for �− to travel forward (backward) with respect to thee− direction. AFB and R, the total cross-section relative to pure QED, are given by

R = F1 , AFB =34

F2

F1, (10.26)

where

F1 = 1 − 2χ0 geV g�

V cos δR + χ20

(ge2V + ge2

A

) (g�2V + g�2

A

), (10.27a)

F2 = −2χ0 geA g�

A cos δR + 4χ20 ge

A g�A ge

V g�V , (10.27b)

tan δR =MZΓZ

M2Z − s

, χ0 =GF

2√

2πα

sM2Z[

(M2Z − s)2 + M2

ZΓ2Z

]1/2, (10.28)

and where√

s is the CM energy. Eq. (10.27) is valid at tree level. If the data areradiatively corrected for QED effects (as described above), then the remaining electroweak

July 30, 2010 14:36


corrections can be incorporated [148,149] (in an approximation adequate for existingPEP, PETRA, and TRISTAN data, which are well below the Z pole) by replacing χ0 byχ(s) ≡ (1 + ρt) χ0(s) α/α(s), where α(s) is the running QED coupling, and evaluating gVin the MS scheme. Reviews and formulae for e+e− → hadrons may be found in Ref. 150.

10.4.2. Z pole physics :At LEP 1 and SLC, there were high-precision measurements of various Z pole

observables [11,151–156], as summarized in Table 10.7. These include the Z mass andtotal width, ΓZ , and partial widths Γ(ff) for Z → ff where fermion f = e, μ, τ , hadrons,b, or c. It is convenient to use the variables MZ , ΓZ , R� ≡ Γ(had)/Γ(�+�−) (� = e, μ, τ),σhad ≡ 12π Γ(e+e−) Γ(had)/M2

Z Γ2Z , Rb ≡ Γ(bb)/Γ(had), and Rc ≡ Γ(cc)/Γ(had), most of

which are weakly correlated experimentally. (Γ(had) is the partial width into hadrons.)The three values for R� are not inconsistent with lepton universality (although Rτ issomewhat low), but we use the general analysis in which the three observables aretreated as independent. Similar remarks apply to A

0,�FB below (A0,τ

FB is somewhat high).O(α3) QED corrections introduce a large anti-correlation (−30%) between ΓZ andσhad. The anti-correlation between Rb and Rc is −18% [11]. The R� are insensitiveto mt except for the Z → bb vertex and final state corrections and the implicitdependence through sin2 θW . Thus, they are especially useful for constraining αs. Thewidth for invisible decays [11], Γ(inv) = ΓZ − 3 Γ(�+�−) − Γ(had) = 499.0 ± 1.5 MeV,can be used to determine the number of neutrino flavors much lighter than MZ/2,Nν = Γ(inv)/Γtheory(νν) = 2.984 ± 0.009 for (mt, MH) = (173.1, 117) GeV.

There were also measurements of various Z pole asymmetries. These include thepolarization or left-right asymmetry

ALR ≡ σL − σR

σL + σR, (10.29)

where σL(σR) is the cross-section for a left-(right-)handed incident electron. ALR wasmeasured precisely by the SLD collaboration at the SLC [152], and has the advantages ofbeing extremely sensitive to sin2 θW and that systematic uncertainties largely cancel. Inaddition, SLD extracted the final-state couplings (defined below), Ab, Ac [11], As [153],Aτ , and Aμ [154], from left-right forward-backward asymmetries, using

AFBLR (f) =

σfLF − σ

fLB − σ

fRF + σ

fRB

σfLF + σ

fLB + σ

fRF + σ

fRB

=34Af , (10.30)

where, for example, σfLF is the cross-section for a left-handed incident electron to

produce a fermion f traveling in the forward hemisphere. Similarly, Aτ was measuredat LEP 1 [11] through the negative total τ polarization, Pτ , and Ae was extractedfrom the angular distribution of Pτ . An equation such as (10.30) assumes that initialstate QED corrections, photon exchange, γ–Z interference, the tiny electroweak boxes,and corrections for

√s �= MZ are removed from the data, leaving the pure electroweak

asymmetries. This allows the use of effective tree-level expressions,

ALR = AePe , AFB =34Af

Ae + Pe

1 + PeAe, (10.31)

July 30, 2010 14:36


where

Af ≡ 2gfV g

fA

gf2V + g

f2A

, (10.32)

andgfV = √

ρf (t(f)3L − 2qfκf sin2 θW ), g

fA = √

ρf t(f)3L . (10.33)

Pe is the initial e− polarization, so that the second equality in Eq. (10.30) is reproducedfor Pe = 1, and the Z pole forward-backward asymmetries at LEP 1 (Pe = 0) aregiven by A

(0,f)FB = 3

4AeAf where f = e, μ, τ , b, c, s [155], and q, and where

A(0,q)FB refers to the hadronic charge asymmetry. Corrections for t-channel exchange and

s/t-channel interference cause A(0,e)FB to be strongly anti-correlated with Re (−37%). The

correlation between A(0,b)FB and A

(0,c)FB amounts to 15%. The initial state coupling, Ae, was

also determined through the left-right charge asymmetry [156] and in polarized Bhabbascattering [154] at the SLC. The forward-backward asymmetry, AFB, for e+e− final states(with invariant masses restricted to or dominated by values around MZ) in pp collisionshas been measured by the CDF [157] and DØ [158] collaborations and values for s2

� wereextracted, which combine to s2

� = 0.2316± 0.0018. By varying the invariant mass and thescattering angle (and assuming the electron couplings), the effective Z couplings to lightquarks, g

u,dV,A, resulted, as well, but with large uncertainties and mutual correlations and

not independently of s2� above. Similar analyses have also been reported by the H1 and

ZEUS collaborations at HERA [159] and by the LEP collaborations [11].The electroweak radiative corrections have been absorbed into corrections ρf − 1 and

κf −1, which depend on the fermion f and on the renormalization scheme. In the on-shellscheme, the quadratic mt dependence is given by ρf ∼ 1 + ρt, κf ∼ 1 + ρt/ tan2 θW ,while in MS, ρf ∼ κf ∼ 1, for f �= b (ρb ∼ 1 − 4

3ρt, κb ∼ 1 + 23ρt). In the MS scheme

the normalization is changed according to GF M2Z/2

√2π → α/4s 2

Z c 2Z . (If one continues

to normalize amplitudes by GF M2Z/2

√2π, as in the 1996 edition of this Review, then

ρf contains an additional factor of ρ.) In practice, additional bosonic and fermionicloops, vertex corrections, leading higher order contributions, etc., must be included.For example, in the MS scheme one has ρ� = 0.9981, κ� = 1.0013, ρb = 0.9870, andκb = 1.0067. It is convenient to define an effective angle s2

f ≡ sin2 θWf ≡ κf s 2Z = κf s2

W ,

in terms of which gfV and gf

A are given by √ρf times their tree-level formulae. Because

g�V is very small, not only A0

LR = Ae, A(0,�)FB , and Pτ , but also A

(0,b)FB , A

(0,c)FB , A

(0,s)FB , and

the hadronic asymmetries are mainly sensitive to s2� . One finds that κf (f �= b) is almost

independent of (mt, MH), so that one can write

s2� ∼ s 2

Z + 0.00029 . (10.34)

Thus, the asymmetries determine values of s2� and s 2

Z almost independent of mt, whilethe κ’s for the other schemes are mt dependent.

July 30, 2010 14:36


10.4.3. LEP 2 :LEP 2 [160] ran at several energies above the Z pole up to ∼ 209 GeV. Measurements

were made of a number of observables, including the cross-sections for e+e− → f ffor f = q, μ−, τ−; the differential cross-sections for f = e−, μ−, τ−; Rq for q = b, c;AFB(f) for f = μ, τ, b, c; W branching ratios; and WW , WWγ, ZZ, single W , andsingle Z cross-sections. They are in good agreement with the SM predictions, with theexceptions of the total hadronic cross-section (1.7 σ high), Rb (2.1 σ low), and AFB(b)(1.6 σ low). Also, the negative result of the direct search for the SM Higgs boson excludedMH values below 114.4 GeV at the 95% CL [161]. This result is complementary to andcan be combined with [162] the limits inferred from the electroweak precision data.

The Z boson properties are extracted assuming the SM expressions for the γ–Zinterference terms. These have also been tested experimentally by performing moregeneral fits [160,163] to the LEP 1 and LEP 2 data. Assuming family universalitythis approach introduces three additional parameters relative to the standard fit [11],describing the γ–Z interference contribution to the total hadronic and leptoniccross-sections, jtot

had and jtot� , and to the leptonic forward-backward asymmetry, jfb

� . E.g.,

jtothad ∼ g�

V ghadV = 0.277 ± 0.065, (10.35)

which is in agreement with the SM expectation [11] of 0.21 ± 0.01. These are valuabletests of the SM; but it should be cautioned that new physics is not expected to bedescribed by this set of parameters, since (i) they do not account for extra interactionsbeyond the standard weak neutral-current, and (ii) the photonic amplitude remains fixedto its SM value.

Strong constraints on anomalous triple and quartic gauge couplings have been obtainedat LEP 2 and the Tevatron as described in the Gauge & Higgs Bosons Particle Listings.

10.4.4. W and Z decays :The partial decay width for gauge bosons to decay into massless fermions f1f2 (the

numerical values include the small electroweak radiative corrections and final state masseffects) is

Γ(W+ → e+νe) =GF M3

W

6√

2π≈ 226.31 ± 0.07 MeV , (10.36a)

Γ(W+ → uidj) =CGF M3

W

6√

2π|Vij |2 ≈ 706.18 ± 0.22 MeV |Vij |2, (10.36b)

Γ(Z → ψiψi) =CGF M3

Z

6√

2π

[gi2V + gi2

A

]

≈

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

167.21 ± 0.02 MeV (νν),83.99 ± 0.01 MeV (e+e−),

300.20 ± 0.06 MeV (uu),382.98 ± 0.06 MeV (dd),375.94 ∓ 0.04 MeV (bb).

(10.36c)

July 30, 2010 14:36


For leptons C = 1, while for quarks

C = 3[1 +

αs(MV )π

+ 1.409α2

s

π2− 12.77

α3s

π3− 80.0

α4s

π4

], (10.37)

where the 3 is due to color and the factor in brackets represents the universal partof the QCD corrections [164] for massless quarks [165]. The O(α4

s) contribution inEq. (10.37) is new [166]. The Z → f f widths contain a number of additional corrections:universal (non-singlet) top quark mass contributions [167]; fermion mass effects andfurther QCD corrections proportional to m2

q(M2Z) [168] which are different for vector

and axial-vector partial widths; and singlet contributions starting from two-loop orderwhich are large, strongly top quark mass dependent, family universal, and flavornon-universal [169]. The QED factor 1 + 3αq2

f/4π, as well as two-loop order ααs andα2 self-energy corrections [170] are also included. Working in the on-shell scheme, i.e.,expressing the widths in terms of GF M3

W,Z , incorporates the largest radiative correctionsfrom the running QED coupling [61,171]. Electroweak corrections to the Z widthsare then incorporated by replacing g i2

V,A by g i2V,A. Hence, in the on-shell scheme the

Z widths are proportional to ρi ∼ 1 + ρt. The MS normalization accounts also forthe leading electroweak corrections [66]. There is additional (negative) quadratic mt

dependence in the Z → bb vertex corrections [172] which causes Γ(bb) to decrease withmt. The dominant effect is to multiply Γ(bb) by the vertex correction 1 + δρbb, where

δρbb ∼ 10−2(− 12

m2t

M2Z

+ 15). In practice, the corrections are included in ρb and κb, as

discussed before.For 3 fermion families the total widths are predicted to be

ΓZ ≈ 2.4957 ± 0.0003 GeV , ΓW ≈ 2.0910 ± 0.0007 GeV . (10.38)

We have assumed αs(MZ) = 0.1200. An uncertainty in αs of ±0.0016 introduces anadditional uncertainty of 0.05% in the hadronic widths, corresponding to ±0.8 MeVin ΓZ . These predictions are to be compared with the experimental results ΓZ =2.4952 ± 0.0023 GeV [11] and ΓW = 2.085 ± 0.042 GeV [173] (see the Gauge & HiggsBoson Particle Listings for more details).

10.5. Precision flavor physics

In addition to cross-sections, asymmetries, parity violation, W and Z decays, thereis a large number of experiments and observables testing the flavor structure of theSM. These are addressed elsewhere in this Review, and generally not included in thisSection. However, we identify three precision observables with sensitivity to similartypes of new physics as the other processes discussed here. The branching fraction ofthe flavor changing transition b → sγ is of comparatively low precision, but since itis a loop-level process (in the SM) its sensitivity to new physics (and SM parameters,such as heavy quark masses) is enhanced. The τ -lepton lifetime and leptonic branching

July 30, 2010 14:36


ratios are primarily sensitive to αs and not affected significantly by many types of newphysics. However, having an independent and reliable low energy measurement of αs

in a global analysis allows the comparison with the Z lineshape determination of αs

which shifts easily in the presence of new physics contributions. By far the most preciseobservable discussed here is the anomalous magnetic moment of the muon (the electronmagnetic moment is measured to even greater precision, but its new physics sensitivity issuppressed by an additional factor of m2

e/m2μ). Its combined experimental and theoretical

uncertainty is comparable to typical new physics contributions.The CLEO [174], BaBar [175] and Belle [176] collaborations reported precise

measurements of the process b → sγ. We extrapolated these results to the full photonspectrum which is defined according to the recommendation in Ref. 177. The results forthe branching fractions are then given by,

CLEO : 3.32 × 10−4[1 ± 0.134 ± 0.076 ± 0.038 ± 0.048 ± 0.003],

BaBar(incl.) : 3.99 × 10−4[1 ± 0.080 ± 0.091 ± 0.079 ± 0.026 ± 0.003],

BaBar(excl.) : 3.57 × 10−4[1 ± 0.055+0.168−0.122 ± 0 ± 0.026 ± 0 ],

Belle : 3.61 × 10−4[1 ± 0.043 ± 0.116 ± 0.003 ± 0.014 ± 0.003],

where the first two errors are the statistical and systematic uncertainties (takenuncorrelated). In the cases of CLEO and Belle, the error component from the signalmodels are moved from the systematic error to the model (third) error. The fourtherror accounts for the extrapolation from the finite photon energy cutoff [177–179](2.0 GeV, 1.9 GeV, and 1.7 GeV, respectively, for CLEO, BaBar and Belle) to the fulltheoretical branching ratio. For this we use the results of Ref. 177 for mb = 4.70 GeVwhich is in good agreement with the more recent Ref. 179. The uncertainty reflects thedifference due to choosing mb = 4.60 GeV, instead. The last error is from the correction(0.957±0.003) for the b → dγ component which is common to all inclusive measurements,but absent for the exclusive BaBar measurement in the third line. The last three errorsare taken as 100% correlated, resulting in the correlation matrix in Table 10.4. It isadvantageous [180] to normalize the result with respect to the semi-leptonic branchingfraction, B(b → Xeν) = 0.1074 ± 0.0016, yielding,

R =B(b → sγ)B(b → Xeν)

= (3.38 ± 0.27 ± 0.37) × 10−3. (10.39)

In the fits we use the variable lnR = −5.69 ± 0.14 to assure an approximately Gaussianerror [181]. The second uncertainty in Eq. (10.39) is an 11% theory uncertainty(excluding parametric errors such as from αs) in the SM prediction which is based onthe next-to-leading order calculations of Refs. 180 and 182. There is a coordinated effortunderway to obtain the SM prediction at next-to-next-to-leading order [183].

The extraction of αs from the τ lifetime is standing out from other determinationsbecause of a variety of independent reasons: (i) the τ -scale is low, so that uponextrapolation to the Z scale (where it can be compared to the theoretically clean

July 30, 2010 14:36


Table 10.4: Correlation matrix for measurements of b → sγ.

CLEO 1.000 0.175 0.048 0.038

BaBar (inclusive) 0.175 1.000 0.029 0.033

BaBar (exclusive) 0.048 0.029 1.000 0.019

Belle 0.038 0.033 0.019 1.000

Z lineshape determinations) the αs error shrinks by about an order of magnitude;(ii) yet, this scale is high enough that perturbation theory and the operator productexpansion (OPE) can be applied; (iii) these observables are fully inclusive and thus freeof fragmentation and hadronization effects that would have to be modeled or measured;(iv) OPE breaking effects are most problematic near the branch cut but there they aresuppressed by a double zero at s = m2

τ ; (v) there are enough data [41,43] to constrainnon-perturbative effects both within and breaking the OPE; (vi) a complete four-looporder QCD calculation is available [166]; (vii) large effects associated with the QCDβ-function can be re-summed [184] in what has become known as contour improvedperturbation theory (CIPT). However, while there is no doubt that CIPT shows fasterconvergence in the lower (calculable) orders, doubts have been cast on the method by theobservation that at least in a specific model [185], which includes the exactly knowncoefficients and theoretical constraints on the large-order behavior, ordinary fixed orderperturbation theory (FOPT) may nevertheless give a better approximation to the fullresult. We therefore use the expressions [5,165,166,186],

ττ =�1 − Bs

τ

Γeτ + Γμ

τ + Γudτ

= 291.09 ± 0.48 fs, (10.40)

Γudτ =

G2F m5

τ |Vud|264π3

S(mτ , MZ)

(1 +

35

m2τ

M2W

)×

[1 +

αs(mτ )π

+ 5.202α2

s

π2+ 26.37

α3s

π3

+ 127.1α4

s

π4+

α

π

(8524

− π2

2

)], (10.41)

and Γeτ and Γμ

τ can be taken from Eq. (10.4) with obvious replacements. The relativefraction of decays with ΔS = −1, Bs

τ = 0.0286 ± 0.0007, is based on experimentaldata since the value for the strange quark mass, ms(mτ ), is not well known andthe QCD expansion proportional to m2

s converges poorly and cannot be trusted.S(mτ , MZ) = 1.01907 ± 0.0003 is a logarithmically enhanced electroweak correctionfactor with higher orders re-summed [187]. Also included (but not shown) are quark

July 30, 2010 14:36


mass effects and condensate contributions [188]. The largest uncertainty arises fromthe truncation of the FOPT series and is conservatively taken as the α4

s term (this isre-calculated in each call of the fits, leading to an αs-dependent and thus asymmetricerror) until a better understanding of the numerical differences between FOPT and CIPThas been gained. Incidentally, the τ spectral functions are better described in CIPTthan in FOPT [188]. Our error almost covers the entire range from using CIPT toassuming that the nearly geometric series in Eq. (10.41) continues to higher orders. Thenext largest uncertainties (±0.6 fs) are from the condensates [188] and the evolutionto the Z scale, followed by the experimental uncertainty in Eq. (10.40), which is fromcombining the two leptonic branching ratios with the direct ττ . Included are also varioussmaller uncertainties from other sources. In total we obtain a ∼ 1.5% determination ofαs(MZ) = 0.1174+0.0018

−0.0016 which updates the result of Ref. 5. For more details, see Refs. 50and 188 where the τ spectral functions are used as additional input.

The world average of the muon anomalous magnetic moment∗∗,

aexpμ =

gμ − 22

= (1165920.80± 0.63) × 10−9, (10.42)

is dominated by the final result of the E821 collaboration at BNL [189]. The QEDcontribution has been calculated to four loops [190] (fully analytically to threeloops [191,192]) , and the leading logarithms are included to five loops [193,194]. Theestimated SM electroweak contribution [195–197], aEW

μ = (1.52 ± 0.03) × 10−9, whichincludes leading two-loop [196] and three-loop [197] corrections, is at the level of twicethe current uncertainty.

The limiting factor in the interpretation of the result is the uncertainty from the two-loop hadronic contribution. E.g., Ref. 50 obtained the value ahad

μ = (69.55± 0.41)× 10−9

which combines CMD-2 [45] and SND [46] e+e− → hadrons cross-section data withradiative return results from BaBar [48] and KLOE [49]. This value suggests a3.1 σ discrepancy between Eq. (10.42) and the SM prediction. Updating an alternativeanalysis [39] the authors of Ref. 40 quote ahad

μ = (70.53 ± 0.45) × 10−9 using τ decaydata and isospin symmetry (CVC). This result implies a smaller conflict (1.8 σ)with Eq. (10.42). Thus, there is also a discrepancy between the 2π and 4π spectralfunctions obtained from the two methods, contributing 70% and 30% of the discrepancy,respectively. For example, if one uses the e+e− data and CVC to predict the branchingratio for τ− → ντπ−π0 decays one obtains 24.78 ± 0.25% [40], while the average of

∗∗ In what follows, we summarize the most important aspects of gμ − 2, and give somedetails about the evaluation in our fits. For more details see the dedicated contribution byA. Hocker and W. Marciano in this Review. There are some small numerical differences (atthe level of 0.1 standard deviation), which are well understood and mostly arise becauseinternal consistency of the fits requires the calculation of all observables from analyticalexpressions and common inputs and fit parameters, so that an independent evaluation isnecessary for this Section. Note, that in the spirit of a global analysis based on all availableinformation we have chosen here to average in the τ decay and radiative return (KLOEand BaBar) data, as well.

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the directly measured branching ratio yields 25.51 ± 0.09% which is 2.7 σ higher. It isimportant to understand the origin of this difference, but two observations point to theconclusion that at least some of it is experimental: (i) The τ− → ντ 2π−π+π0 spectralfunction also disagrees with the corresponding e+e− data by 1.7 σ, which translates to a20% effect [40,50] and seems too large to arise from isospin violation. (ii) Isospin violatingcorrections have been studied in detail in Refs. 40 and 198 and found to be largelyunder control. The largest effect is due to higher-order electroweak corrections [52] butintroduces a negligible uncertainty [187]. Nevertheless, ahad

μ is often evaluated excludingthe τ decay data arguing [199] that CVC breaking effects (e.g., through a relatively largemass difference between the ρ± and ρ0 vector mesons) may be larger than expected.(This may also be relevant [199] in the context of the NuTeV result discussed above.)Experimentally [43], this mass difference is indeed larger than expected, but then onewould also expect a significant width difference which is contrary to observation [43].Fortunately, due to the suppression at large s (from where the conflicts originate) theseproblems are less pronounced as far as ahad

μ is concerned. In the following we view alldifferences in spectral functions as (systematic) fluctuations and average the results.

Table 10.5: Principal SM fit result including mutualcorrelations (all masses in GeV).

MZ 91.1874 ± 0.0021 1.00 −0.01 0.00 0.00 −0.01 0.00 0.12

mt(mt) 163.5 ± 1.3 −0.01 1.00 0.00 0.00 −0.10 0.00 0.39

mb(mb) 4.198 ± 0.023 0.00 0.00 1.00 0.25 −0.04 0.01 0.04

mc(mc) 1.266+0.031−0.036 0.00 0.00 0.25 1.00 0.08 0.02 0.12

αs(MZ) 0.1183 ± 0.0015 −0.01 −0.10 −0.04 0.08 1.00 0.00 −0.04

Δα(3)had(1.8 GeV) 0.00574 ± 0.00010 0.00 −0.01 0.01 0.02 0.00 1.00 −0.18

MH 90+27−22 0.12 0.39 0.04 0.12 −0.04 −0.18 1.00

An additional uncertainty is induced by the hadronic three-loop light-by-lightscattering contribution. Two recent and inherently different model calculations yieldaLBLSμ = (+1.36 ± 0.25) × 10−9 [200] and aLBLS

μ = +1.37+0.15−0.27 × 10−9 [201] which are

higher than previous evaluations [202,203]. The sign of this effect is opposite [202] tothe one quoted in the 2002 edition of this Review, and has subsequently been confirmedby two other groups [203]. There is also the upper bound aLBLS

μ < 1.59 × 10−9 [201]but this requires an ad hoc assumption, too. The very recent Ref. 204 quotes the valueaLBLSμ = (+1.05 ± 0.26) × 10−9, which we shift by 2 × 10−11 to account for the more

accurate charm quark treatment of Ref. 201. We also increase the error to cover allevaluations, and we will use aLBLS

μ = (+1.07 ± 0.32) × 10−9 in the fits.

Other hadronic effects at three-loop order contribute [205] ahadμ (α3) = (−1.00 ±

July 30, 2010 14:36


0.06) × 10−9. Correlations with the two-loop hadronic contribution and with Δα(MZ)(see Sec. 10.2) were considered in Ref. 192 which also contains analytic results for theperturbative QCD contribution.

Altogether, the SM prediction is

atheoryμ = (1165918.90± 0.44) × 10−9 , (10.43)

where the error is from the hadronic uncertainties excluding parametric ones such as fromαs and the heavy quark masses. We estimate its correlation with Δα(MZ) to 14%. Theoverall 2.5 σ discrepancy between the experimental and theoretical values could be due tofluctuations (the E821 result is statistics dominated) or underestimates of the theoreticaluncertainties. On the other hand, gμ − 2 is also affected by many types of new physics,such as supersymmetric models with large tanβ and moderately light superparticlemasses [206]. Thus, the deviation could also arise from physics beyond the SM.

10.6. Experimental results

The values for mt [6], MW [160,207], neutrino scattering [98,114–116], theweak charges of the electron [125], cesium [132,133] and thallium [134], the b → sγobservable [174–175], the muon anomalous magnetic moment [189], and the τ lifetimeare listed in Table 10.6. Likewise, the principal Z pole observables can be found inTable 10.7 where the LEP 1 averages of the ALEPH, DELPHI, L3, and OPAL resultsinclude common systematic errors and correlations [11]. The heavy flavor results ofLEP 1 and SLD are based on common inputs and correlated, as well [11]. Note that thevalues of Γ(�+�−), Γ(had), and Γ(inv) are not independent of ΓZ , the R�, and σhad andthat the SM errors in those latter are largely dominated by the uncertainty in αs. Alsoshown in both Tables are the SM predictions for the values of MZ , MH , αs(MZ), Δα

(3)had

and the heavy quark masses shown in Table 10.5. The predictions result from a globalleast-square (χ2) fit to all data using the minimization package MINUIT [208] and theelectroweak library GAPP [90]. In most cases, we treat all input errors (the uncertaintiesof the values) as Gaussian. The reason is not that we assume that theoretical andsystematic errors are intrinsically bell-shaped (which they are not) but because in mostcases the input errors are combinations of many different (including statistical) errorsources, which should yield approximately Gaussian combined errors by the large numbertheorem. Thus, it suffices if either the statistical components dominate or there aremany components of similar size. An exception is the theory dominated error on theτ lifetime, which we recalculate in each χ2-function call since it depends itself on αs.Sizes and shapes of the output errors (the uncertainties of the predictions and the SM fitparameters) are fully determined by the fit, and 1σ errors are defined to correspond toΔχ2 = χ2 − χ2

min = 1, and do not necessarily correspond to the 68.3% probability rangeor the 39.3% probability contour (for 2 parameters).

The agreement is generally very good. Despite the few discrepancies discussed in thefollowing, the fit describes well the data with a χ2/d.o.f. = 43.0/44. Only the final resultfor gμ − 2 from BNL and A

(0,b)FB from LEP 1 are currently showing large (2.5 σ and 2.7 σ)

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Table 10.6: Principal non-Z pole observables, compared with the SM best fitpredictions. The first MW value is from the Tevatron [207] and the second onefrom LEP 2 [160]. The values of MW and mt differ from those in the ParticleListings when they include recent preliminary results. g2

L, which has been adjustedas discussed in Sec. 10.3, and g2

R are from NuTeV [98] and have a very small(−1.7%) residual anti-correlation. e-DIS [123] and the older ν-DIS constraints fromCDHS [92], CHARM [93], and CCFR [94] are included, as well, but not shownin the Table. The world averages for gνe

V,A are dominated by the CHARM II [116]results, gνe

V = −0.035 ± 0.017 and gνeA = −0.503 ± 0.017. The errors are the total

(experimental plus theoretical) uncertainties. The ττ value is the τ lifetime worldaverage computed by combining the direct measurements with values derived fromthe leptonic branching ratios [5]; in this case, the theory uncertainty is included inthe SM prediction. In all other SM predictions, the uncertainty is from MZ , MH ,mt, mb, mc, α(MZ), and αs, and their correlations have been accounted for. Thecolumn denoted Pull gives the standard deviations for the principal fit with MH

free, while the column denoted Dev. (Deviation) is for MH = 117 GeV fixed.

Quantity Value Standard Model Pull Dev.

mt [GeV] 173.1 ± 1.3 173.2 ± 1.3 −0.1 −0.5MW [GeV] 80.420 ± 0.031 80.384 ± 0.014 1.2 1.5

80.376 ± 0.033 −0.2 0.1g2L 0.3027 ± 0.0018 0.30399 ± 0.00017 −0.7 −0.6

g2R 0.0308 ± 0.0011 0.03001 ± 0.00002 0.7 0.7

gνeV −0.040 ± 0.015 −0.0398 ± 0.0003 0.0 0.0

gνeA −0.507 ± 0.014 −0.5064 ± 0.0001 0.0 0.0

QW (e) −0.0403 ± 0.0053 −0.0473 ± 0.0005 1.3 1.2QW (Cs) −73.20 ± 0.35 −73.15 ± 0.02 −0.1 −0.1QW (Tl) −116.4 ± 3.6 −116.76 ± 0.04 0.1 0.1ττ [fs] 291.09 ± 0.48 290.02± 2.09 0.5 0.5Γ(b→sγ)

Γ(b→Xeν)

(3.38+0.51

−0.44

)× 10−3 (3.11 ± 0.07) × 10−3 0.6 0.6

12 (gμ − 2 − α

π ) (4511.07± 0.77) × 10−9 (4509.13± 0.08) × 10−9 2.5 2.5

deviations. In addition, A0LR (SLD) from hadronic final states differs by 1.8 σ. The SM

prediction of σhad (LEP 1) moved closer to the measurement value which is slightlyhigher. Rb, whose measured value deviated in the past by as much as 3.7 σ from the SMprediction, is now in agreement, and a 2 σ discrepancy in QW (Cs) has also been resolved.g2L from NuTeV is currently in agreement with the SM but this statement is preliminary

(see Sec. 10.3).

Ab can be extracted from A(0,b)FB when Ae = 0.1501 ± 0.0016 is taken from a fit to

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Table 10.7: Principal Z pole observables and their SM predictions (cf. Table 10.6).The first s2

� (A(0,q)FB ) is the effective angle extracted from the hadronic charge

asymmetry while the second is the combined lepton asymmetry from CDF [157] andDØ [158]. The three values of Ae are (i) from ALR for hadronic final states [152];(ii) from ALR for leptonic final states and from polarized Bhabba scattering [154];and (iii) from the angular distribution of the τ polarization at LEP 1. The two Aτ

values are from SLD and the total τ polarization, respectively.

Quantity Value Standard Model Pull Dev.

MZ [GeV] 91.1876 ± 0.0021 91.1874 ± 0.0021 0.1 0.0ΓZ [GeV] 2.4952 ± 0.0023 2.4954 ± 0.0009 −0.1 0.1Γ(had) [GeV] 1.7444 ± 0.0020 1.7418 ± 0.0009 — —Γ(inv) [MeV] 499.0 ± 1.5 501.69 ± 0.07 — —Γ(�+�−) [MeV] 83.984 ± 0.086 84.005 ± 0.015 — —σhad[nb] 41.541 ± 0.037 41.484 ± 0.008 1.5 1.5Re 20.804 ± 0.050 20.735 ± 0.010 1.4 1.4Rμ 20.785 ± 0.033 20.735 ± 0.010 1.5 1.6Rτ 20.764 ± 0.045 20.780 ± 0.010 −0.4 −0.3Rb 0.21629 ± 0.00066 0.21578 ± 0.00005 0.8 0.8Rc 0.1721 ± 0.0030 0.17224 ± 0.00003 0.0 0.0

A(0,e)FB 0.0145 ± 0.0025 0.01633 ± 0.00021 −0.7 −0.7

A(0,μ)FB 0.0169 ± 0.0013 0.4 0.6

A(0,τ)FB 0.0188 ± 0.0017 1.5 1.6

A(0,b)FB 0.0992 ± 0.0016 0.1034 ± 0.0007 −2.7 −2.3

A(0,c)FB 0.0707 ± 0.0035 0.0739 ± 0.0005 −0.9 −0.8

A(0,s)FB 0.0976 ± 0.0114 0.1035 ± 0.0007 −0.6 −0.4

s2� (A

(0,q)FB ) 0.2324 ± 0.0012 0.23146 ± 0.00012 0.8 0.7

0.2316 ± 0.0018 0.1 0.0Ae 0.15138 ± 0.00216 0.1475 ± 0.0010 1.8 2.2

0.1544 ± 0.0060 1.1 1.30.1498 ± 0.0049 0.5 0.6

Aμ 0.142 ± 0.015 −0.4 −0.3Aτ 0.136 ± 0.015 −0.8 −0.7

0.1439 ± 0.0043 −0.8 −0.7Ab 0.923 ± 0.020 0.9348 ± 0.0001 −0.6 −0.6Ac 0.670 ± 0.027 0.6680 ± 0.0004 0.1 0.1As 0.895 ± 0.091 0.9357 ± 0.0001 −0.4 −0.4

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leptonic asymmetries (using lepton universality). The result, Ab = 0.881 ± 0.017, is 3.2 σbelow the SM prediction† and also 1.6 σ below Ab = 0.923± 0.020 obtained from AFB

LR (b)

at SLD. Thus, it appears that at least some of the problem in A(0,b)FB is experimental.

Note, however, that the uncertainty in A(0,b)FB is strongly statistics dominated. The

combined value, Ab = 0.899± 0.013 deviates by 2.8 σ. It would be difficult to account forthis 4.0% deviation by new physics that enters only at the level of radiative correctionssince about a 20% correction to κb would be necessary to account for the central valueof Ab [211]. If this deviation is due to new physics, it is most likely of tree-level typeaffecting preferentially the third generation. Examples include the decay of a scalarneutrino resonance [209], mixing of the b quark with heavy exotics [210], and a heavy Z ′with family-nonuniversal couplings [212,213]. It is difficult, however, to simultaneouslyaccount for Rb, which has been measured on the Z peak and off-peak [214] at LEP 1.An average of Rb measurements at LEP 2 at energies between 133 and 207 GeV is 2.1 σ

below the SM prediction, while A(b)FB (LEP 2) is 1.6 σ low [160].

The left-right asymmetry, A0LR = 0.15138 ± 0.00216 [152], based on all hadronic data

from 1992–1998 differs 1.8 σ from the SM expectation of 0.1475 ± 0.0010. The combinedvalue of A� = 0.1513 ± 0.0021 from SLD (using lepton-family universality and includingcorrelations) is also 1.8 σ above the SM prediction; but there is now experimentalagreement between this SLD value and the LEP 1 value, A� = 0.1481 ± 0.0027, obtainedfrom a fit to A

(0,�)FB , Ae(Pτ ), and Aτ (Pτ ), again assuming universality.

The observables in Table 10.6 and Table 10.7, as well as some other less preciseobservables, are used in the global fits described below. In all fits, the errors includefull statistical, systematic, and theoretical uncertainties. The correlations on the LEP 1lineshape and τ polarization, the LEP/SLD heavy flavor observables, the SLD leptonasymmetries, and the deep inelastic and ν-e scattering observables, are included. Thetheoretical correlations between Δα

(5)had and gμ − 2, and between the charm and bottom

quark masses, are also accounted for.The data allow a simultaneous determination of MZ , MH , mt, and the strong coupling

αs(MZ). (mc, mb, and Δα(3)had are also allowed to float in the fits, subject to the

theoretical constraints [5,17] described in Sec. 10.1–Sec. 10.2. These are correlated withαs.) αs is determined mainly from R�, ΓZ , σhad, and ττ and is only weakly correlatedwith the other variables. The global fit to all data, including the CDF/DØ averagemt = 173.1 ± 1.3 GeV, yields the result in Table 10.5 (the MS top quark mass given therecorresponds to mt = 173.2 ± 1.3 GeV). The weak mixing angle is determined to

s 2Z = 0.23116 ± 0.00013, s2

W = 0.22292± 0.00028,

where the larger error in the on-shell scheme is due to the stronger sensitivity to mt, whilethe corresponding effective angle is related by Eq. (10.34), i.e., s2

� = 0.23146 ± 0.00012.

† Alternatively, one can use A� = 0.1481 ± 0.0027, which is from LEP 1 alone and inexcellent agreement with the SM, and obtain Ab = 0.893± 0.022 which is 1.9 σ low. Thisillustrates that some of the discrepancy is related to the one in ALR.

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As described at the beginning of Sec. 10.2 and the paragraph following Eq. (10.42)in Sec. 10.5, there is considerable stress in the experimental e+e− spectral functionsand also conflict when these are compared with τ decay spectral functions. These arebelow or above the 2σ level (depending on what is actually compared) but not largerthan the deviations of some other quantities entering our analyzes. The number andsize or these deviations are not inconsistent with what one would expect to happenas a result of random fluctuations. It is nevertheless instructive to study the effect ofdoubling the uncertainty in Δα

(3)had(1.8 GeV) = (57.29± 0.90) × 10−4, (see the beginning

of Sec. 10.2) on the extracted Higgs mass. The result, MH = 87+28−22 GeV, demonstrates

that the uncertainty in Δαhad is currently of only secondary importance. Note also, thatthe uncertainty of about ±0.0001 in Δα

(3)had(1.8 GeV) corresponds to a shift of ∓5 GeV

in MH or about one fifth of its total uncertainty. The hadronic contribution to α(MZ)is correlated with gμ − 2 (see Sec. 10.5). The measurement of the latter is higher thanthe SM prediction, and its inclusion in the fit favors a larger α(MZ) and a lower MH

(currently by about 2 GeV).

The weak mixing angle can be determined from Z pole observables, MW , and froma variety of neutral-current processes spanning a very wide Q2 range. The results (forthe older low energy neutral-current data see Refs. 58 and 59) shown in Table 10.8are in reasonable agreement with each other, indicating the quantitative success of theSM. The largest discrepancy is the value s 2

Z = 0.23193 ± 0.00028 from the forward-backward asymmetries into bottom and charm quarks, which is 2.7 σ above the value0.23116 ± 0.00013 from the global fit to all data. Similarly, s 2

Z = 0.23067 ± 0.00029from the SLD asymmetries (in both cases when combined with MZ) is 1.7 σ low.The SLD result has the additional difficulty (within the SM) of implying very low andexcluded [161] Higgs masses. This is also true for s 2

Z = 0.23100 ± 0.00023 from MWand MZ and — as a consequence — for the global fit. We have therefore included inTable 10.6 and Table 10.7 an additional column (denoted Deviation) indicating thedeviations if MH = 117 GeV is fixed.

The extracted Z pole value of αs(MZ) is based on a formula with negligible theoreticaluncertainty if one assumes the exact validity of the SM. One should keep in mind,however, that this value, αs(MZ) = 0.1198± 0.0028, is very sensitive to such types of newphysics as non-universal vertex corrections. In contrast, the value derived from τ decays,αs(MZ) = 0.1174+0.0018

−0.0016, is theory dominated but less sensitive to new physics. The twovalues are in remarkable agreement with each other. They are also in agreement withother recent values, such as from jet-event shapes at LEP [215] (0.1202 ± 0.0050) theaverage from HERA [216] (0.1198 ± 0.0032), and the most recent unquenched latticecalculation [217] (0.1183 ± 0.0008). For more details and other determinations, see ourSection 9 on “Quantum Chromodynamics” in this Review.

Using α(MZ) and s 2Z as inputs, one can predict αs(MZ) assuming grand unification.

One predicts [218] αs(MZ) = 0.130 ± 0.001 ± 0.01 for the simplest theories based on theminimal supersymmetric extension of the SM, where the first (second) uncertainty isfrom the inputs (thresholds). This is slightly larger, but consistent with the experimentalαs(MZ) = 0.1183 ± 0.0015 from the Z lineshape and the τ lifetime, as well as

July 30, 2010 14:36


Table 10.8: Values of s 2Z , s2

W , αs, and MH [in GeV] for various (combinations of)observables. Unless indicated otherwise, the top quark mass, mt = 170.9± 1.9 GeV,is used as an additional constraint in the fits. The (†) symbol indicates a fixedparameter.

Data s 2Z s2

W αs(MZ) MH

All data 0.23116(13) 0.22292(28) 0.1183(15) 90+27−22

All indirect (no mt) 0.23118(14) 0.22283(34) 0.1183(16) 112+110− 52

Z pole (no mt) 0.23121(17) 0.22311(59) 0.1198(28) 90+114− 44

LEP 1 (no mt) 0.23152(21) 0.22376(67) 0.1213(30) 170+234− 93

SLD + MZ 0.23067(29) 0.22201(54) 0.1183 (†) 33+27−17

A(b,c)FB + MZ 0.23193(28) 0.22484(76) 0.1183 (†) 389+264

−158

MW + MZ 0.23100(23) 0.22262(48) 0.1183 (†) 67+38−28

MZ 0.23128(6) 0.22321(17) 0.1183 (†) 117 (†)QW (APV) 0.2314(14) 0.2233(14) 0.1183 (†) 117 (†)QW (e) 0.2332(15) 0.2251(15) 0.1183 (†) 117 (†)νμ-N DIS (isoscalar) 0.2335(18) 0.2254(18) 0.1183 (†) 117 (†)Elastic νμ(νμ)-e 0.2311(77) 0.2230(77) 0.1183 (†) 117 (†)e-D DIS (SLAC) 0.222(18) 0.213(19) 0.1183 (†) 117 (†)Elastic νμ(νμ)-p 0.211(33) 0.203(33) 0.1183 (†) 117 (†)

with other determinations. Non-supersymmetric unified theories predict the low valueαs(MZ) = 0.073 ± 0.001 ± 0.001. See also the note on “Supersymmetry” in the SearchesParticle Listings.

The data indicate a preference for a small Higgs mass. There is a strong correlationbetween the quadratic mt and logarithmic MH terms in ρ in all of the indirect dataexcept for the Z → bb vertex. Therefore, observables (other than Rb) which favor mt

values higher than the Tevatron range favor lower values of MH . MW has additional MH

dependence through ΔrW which is not coupled to m2t effects. The strongest individual

pulls toward smaller MH are from MW and A0LR, while A

(0,b)FB favors higher values. The

difference in χ2 for the global fit is Δχ2 = χ2(MH = 300 GeV) − χ2min ≈ 25. Hence,

the data favor a small value of MH , as in supersymmetric extensions of the SM. Thecentral value of the global fit result, MH = 90+27

−22 GeV, is below the direct lower bound,MH ≥ 114.4 GeV (95% CL) [161].

July 30, 2010 14:36


The 90% central confidence range from all precision data is

55 GeV ≤ MH ≤ 135 GeV. (10.44)

Including the results of the direct searches at LEP 2 [161] and the Tevatron [219] as extracontributions to the likelihood function drives the 95% upper limit to MH ≤ 147 GeV.As two further refinements, we account for (i) theoretical uncertainties from uncalculatedhigher order contributions by allowing the T parameter (see next subsection) subject tothe constraint T = 0 ± 0.02, (ii) the MH dependence of the correlation matrix whichgives slightly more weight to lower Higgs masses [220]. The resulting limits at 95 (90,99)% CL are, respectively,

MH ≤ 149 (145, 194) GeV. (10.45)

One can also carry out a fit to the indirect data alone, i.e., without including theconstraint, mt = 173.1 ± 1.3 GeV, obtained by CDF and DØ. (The indirect predictionis for the MS mass, mt(mt) = 166.2+8.1

−6.6 GeV, which is in the end converted to the polemass). One obtains mt = 176.0+8.5

−7.0 GeV, in perfect agreement with the direct CDF/DØaverage. Using this indirect top mass value, the tendency for a light Higgs persists andEq. (10.44) becomes 46 GeV ≤ MH ≤ 306 GeV. The relations between MH and mt forvarious observables are shown in Fig. 10.2.

One can also determine the radiative correction parameters Δr: from the global fit oneobtains Δr = 0.0351±0.0009 and ΔrW = 0.06950±0.00021. MW measurements [160,207](when combined with MZ) are equivalent to measurements of Δr = 0.0342 ± 0.0014,which is 0.9 σ below the result from all other data, Δr = 0.0357 ± 0.0011. Fig. 10.3shows the 1 σ contours in the MW -mt plane from the direct and indirect determinations,as well as the combined 90% CL region. The indirect determination uses MZ fromLEP 1 as input, which is defined assuming an s-dependent decay width. MW thencorresponds to the s-dependent width definition, as well, and can be directly comparedwith the results from the Tevatron and LEP 2 which have been obtained using the samedefinition. The difference to a constant width definition is formally only of O(α2), butis strongly enhanced since the decay channels add up coherently. It is about 34 MeVfor MZ and 27 MeV for MW . The residual difference between working consistently withone or the other definition is about 3 MeV, i.e., of typical size for non-enhanced O(α2)corrections [72–75].

Most of the parameters relevant to ν-hadron, ν-e, e-hadron, and e−e± processes aredetermined uniquely and precisely from the data in “model-independent” fits (i.e., fitswhich allow for an arbitrary electroweak gauge theory). The values for the parametersdefined in Eqs. (10.11)–(10.14) are given in Table 10.9 along with the predictions of theSM. The agreement is very good. (The ν-hadron results without the NuTeV data can befound in the 1998 edition of this Review, and the fits using the original NuTeV data in the2006 edition.) The off Z pole e+e− results are difficult to present in a model-independentway because Z propagator effects are non-negligible at TRISTAN, PETRA, PEP, andLEP 2 energies. However, assuming e-μ-τ universality, the low energy lepton asymmetriesimply [150] 4 (ge

A)2 = 0.99 ± 0.05, in good agreement with the SM prediction � 1.

July 30, 2010 14:36


145 150 155 160 165 170 175 180 185 190mt [GeV]

10

20

30

50

100

200

300

500

1000

MH [

GeV

]

LEP 2

Tevatron excluded (95% CL)

excluded

all data (90% CL)

(95% CL)

ΓZ, σ

had, R

l, R

q

Z pole asymmetriesMW

low energymt

Figure 10.2: One-standard-deviation (39.35%) uncertainties in MH as a functionof mt for various inputs, and the 90% CL region (Δχ2 = 4.605) allowed by all data.αs(MZ) = 0.1183 is assumed except for the fits including the Z lineshape or lowenergy data. The direct lower limit from LEP 2 and the excluded window from theTevatron [221] (both at the 95% CL) are also shown. Color version at end of book.

10.7. Constraints on new physics

The Z pole, W mass, and low energy data can be used to search for and set limits ondeviations from the SM. In particular, the combination of these indirect data with thedirect CDF and DØ average for mt allows one to set stringent limits on new physics. Wewill mainly discuss the effects of exotic particles (with heavy masses Mnew MZ in anexpansion in MZ/Mnew) on the gauge boson self-energies. (Brief remarks are made onnew physics which is not of this type.) Most of the effects on precision measurements canbe described by three gauge self-energy parameters S, T , and U . We will define these, aswell as related parameters, such as ρ0, εi, and εi, to arise from new physics only. I.e.,they are equal to zero (ρ0 = 1) exactly in the SM, and do not include any contributionsfrom mt or MH , which are treated separately. Our treatment differs from most of theoriginal papers.

Many extensions of the SM can be described by the ρ0 parameter,

ρ0 ≡ M2W

M2Z c 2

Z ρ, (10.46)

which describes new sources of SU(2) breaking that cannot be accounted for by the SM

July 30, 2010 14:36


160 165 170 175 180 185mt [GeV]

80.3

80.35

80.4

80.45

MW

[G

eV]

M H = 117 G

eV

M H = 200 G

eV

M H = 300 G

eV

M H = 500 G

eV

direct (1σ)

indirect (1σ)

all data (90%)

Figure 10.3: One-standard-deviation (39.35%) region in MW as a function of mt

for the direct and indirect data, and the 90% CL region (Δχ2 = 4.605) allowed byall data. The SM prediction as a function of MH is also indicated. The widths ofthe MH bands reflect the theoretical uncertainty from α(MZ). Color version at endof book.

Higgs doublet or mt effects. In the presence of ρ0 �= 1, Eq. (10.46) generalizes the secondEq. (10.8) while the first remains unchanged. Provided that the new physics whichyields ρ0 �= 1 is a small perturbation which does not significantly affect the radiativecorrections, ρ0 can be regarded as a phenomenological parameter which multiplies GF

in Eqs. (10.11)–(10.14), (10.28), and ΓZ in Eq. (10.36c). There are enough data todetermine ρ0, MH , mt, and αs, simultaneously. From the global fit,

ρ0 = 1.0008+0.0017−0.0007 , (10.47)

114.4 GeV ≤ MH ≤ 427 GeV, (10.48)

mt = 173.1 ± 1.3 GeV, (10.49)

αs(MZ) = 0.1181 ± 0.0015, (10.50)

where the lower limit on MH is the direct search bound. (If the direct limit is ignoredone obtains MH = 162+265

− 93 GeV and ρ0 = 1.0008+0.0017−0.0010.) The error bar in Eq. (10.47)

is highly asymmetric: at the 2 σ level one has ρ0 = 1.0004+0.0029−0.0011 with no meaningful

bound on MH . The result in Eq. (10.47) is slightly above but consistent with the SMexpectation, ρ0 = 1. It can be used to constrain higher-dimensional Higgs representationsto have vacuum expectation values of less than a few percent of those of the doublets.Indeed, the relation between MW and MZ is modified if there are Higgs multiplets with

July 30, 2010 14:36


Table 10.9: Values of the model-independent neutral-current parameters, comparedwith the SM predictions. There is a second gνe

V,A solution, given approximatelyby gνe

V ↔ gνeA , which is eliminated by e+e− data under the assumption that the

neutral current is dominated by the exchange of a single Z boson. The εL, as wellas the εR, are strongly correlated and non-Gaussian, so that for implementations werecommend the parametrization using g2

i and θi = tan−1[εi(u)/εi(d)], i = L or R.The analysis of more recent low energy experiments in polarized electron scatteringperformed in Ref. 123 is included by means of the two orthogonal constraints,cos γ C1d − sin γ C1u = 0.342 ± 0.063 and sin γ C1d + cos γ C1u = −0.0285 ± 0.0043,where tan γ ≈ 0.445. In the SM predictions, the uncertainty is from MZ , MH , mt,mb, mc, α(MZ), and αs.

ExperimentalQuantity Value SM Correlation

εL(u) 0.338 ±0.016 0.3461(1)εL(d) −0.434 ±0.012 −0.4292(1) non-εR(u) −0.174 +0.013

−0.004 −0.1549(1) GaussianεR(d) −0.023 +0.071

−0.047 0.0775

g2L 0.3025±0.0014 0.3040(2) −0.18 −0.21 −0.02

g2R 0.0309±0.0010 0.0300 −0.03 −0.07

θL 2.48 ±0.036 2.4630(1) 0.24θR 4.58 +0.41

−0.28 5.1765

gνeV −0.040 ±0.015 −0.0398(3) −0.05

gνeA −0.507 ±0.014 −0.5064(1)

C1u + C1d 0.1537 ±0.0011 0.1528(1) 0.64 −0.18 −0.01C1u − C1d −0.516 ±0.014 −0.5300(3) −0.27 −0.02C2u + C2d −0.21 ±0.57 −0.0089 −0.30C2u − C2d −0.077 ±0.044 −0.0625(5)

QW (e) = −2C2e −0.0403±0.0053 −0.0473(5)

weak isospin > 1/2 with significant vacuum expectation values. In order to calculate tohigher orders in such theories one must define a set of four fundamental renormalizedparameters which one may conveniently choose to be α, GF , MZ , and MW , sinceMW and MZ are directly measurable. Then s 2

Z and ρ0 can be considered dependentparameters.

Eq. (10.47) can also be used to constrain other types of new physics. For example,

July 30, 2010 14:36


non-degenerate multiplets of heavy fermions or scalars break the vector part of weakSU(2) and lead to a decrease in the value of MZ/MW . A non-degenerate SU(2) doublet(f1f2

)yields a positive contribution to ρ0 [222] of

C GF

8√

2π2Δm2, (10.51)

where

Δm2 ≡ m21 + m2

2 − 4m21m

22

m21 − m2

2

lnm1

m2≥ (m1 − m2)2, (10.52)

and C = 1 (3) for color singlets (triplets). Thus, in the presence of such multiplets,

ρ0 = 1 +3 GF

8√

2π2

∑i

Ci

3Δm2

i , (10.53)

where the sum includes fourth-family quark or lepton doublets,(t′b′)

or(E0

E−), and scalar

doublets such as(tb

)in Supersymmetry (in the absence of L–R mixing). This implies

∑i

Ci

3Δm2

i ≤ (106 GeV)2 (10.54)

at 95% CL. The corresponding constraints on non-degenerate squark and slepton doubletsare even stronger,

∑i CiΔm2

i /3 ≤ (61 GeV)2. This is due to the supersymmetric Higgsmass bound, mh0 < 150 GeV, and the very strong correlation between mh0 and ρ0 (97%).

Non-degenerate multiplets usually imply ρ0 > 1. Similarly, heavy Z ′ bosons decreasethe prediction for MZ due to mixing and generally lead to ρ0 > 1 [223]. On theother hand, additional Higgs doublets which participate in spontaneous symmetrybreaking [224], heavy lepton doublets involving Majorana neutrinos [225], and thevacuum expectation values of Higgs triplets or higher-dimensional representations cancontribute to ρ0 with either sign. Allowing for the presence of heavy degenerate chiralmultiplets (the S parameter, to be discussed below) affects the determination of ρ0 fromthe data, at present leading to a larger value (for fixed MH).

A number of authors [226–231] have considered the general effects on neutral-currentand Z and W boson observables of various types of heavy (i.e., Mnew MZ) physicswhich contribute to the W and Z self-energies but which do not have any direct couplingto the ordinary fermions. In addition to non-degenerate multiplets, which break thevector part of weak SU(2), these include heavy degenerate multiplets of chiral fermionswhich break the axial generators. The effects of one degenerate chiral doublet are small,but in Technicolor theories there may be many chiral doublets and therefore significanteffects [226].

Such effects can be described by just three parameters, S, T , and U , at the(electroweak) one-loop level. (Three additional parameters are needed if the new physics

July 30, 2010 14:36


scale is comparable to MZ [232]. Further generalizations, including effects relevant toLEP 2, are described in Ref. 233.) T is proportional to the difference between the Wand Z self-energies at Q2 = 0 (i.e., vector SU(2)-breaking), while S (S + U) is associatedwith the difference between the Z (W ) self-energy at Q2 = M2

Z,W and Q2 = 0 (axialSU(2)-breaking). Denoting the contributions of new physics to the various self-energiesby Πnew

ij , we have

α(MZ)T ≡ ΠnewWW (0)M2

W

− ΠnewZZ (0)M2

Z

, (10.55a)

α(MZ)4 s 2

Z c 2Z

S ≡ ΠnewZZ (M2

Z) − ΠnewZZ (0)

M2Z

−

c 2Z − s 2

Z

c Z s Z

ΠnewZγ (M2

Z)

M2Z

− Πnewγγ (M2

Z)

M2Z

, (10.55b)

α(MZ)4 s 2

Z

(S + U) ≡ ΠnewWW (M2

W ) − ΠnewWW (0)

M2W

−

c Z

s Z

ΠnewZγ (M2

Z)

M2Z

− Πnewγγ (M2

Z)

M2Z

. (10.55c)

S, T , and U are defined with a factor proportional to α removed, so that they areexpected to be of order unity in the presence of new physics. In the MS scheme as definedin Ref. 64, the last two terms in Eqs. (10.55b) and (10.55c) can be omitted (as was donein some earlier editions of this Review). These three parameters are related to otherparameters (Si, hi, εi) defined in Refs. [64,227,228] by

T = hV = ε1/α(MZ),

S = hAZ = SZ = 4 s 2Z ε3/α(MZ),

U = hAW − hAZ = SW − SZ = −4 s 2Z ε2/α(MZ). (10.56)

A heavy non-degenerate multiplet of fermions or scalars contributes positively to T as

ρ0 − 1 =1

1 − α(MZ)T− 1 � α(MZ)T, (10.57)

where ρ0 is given in Eq. (10.53). The effects of non-standard Higgs representations cannotbe separated from heavy non-degenerate multiplets unless the new physics has otherconsequences, such as vertex corrections. Most of the original papers defined T to includethe effects of loops only. However, we will redefine T to include all new sources of SU(2)breaking, including non-standard Higgs, so that T and ρ0 are equivalent by Eq. (10.57).

July 30, 2010 14:36


A multiplet of heavy degenerate chiral fermions yields

S =C

3π

∑i

(t3L(i) − t3R(i)

)2, (10.58)

where t3L,R(i) is the third component of weak isospin of the left-(right-)handedcomponent of fermion i and C is the number of colors. For example, a heavy degenerateordinary or mirror family would contribute 2/3π to S. In Technicolor models withQCD-like dynamics, one expects [226] S ∼ 0.45 for an iso-doublet of techni-fermions,assuming NTC = 4 techni-colors, while S ∼ 1.62 for a full techni-generation withNTC = 4; T is harder to estimate because it is model-dependent. In these examplesone has S ≥ 0. However, the QCD-like models are excluded on other grounds (flavorchanging neutral-currents, and too-light quarks and pseudo-Goldstone bosons [234]) .In particular, these estimates do not apply to models of walking Technicolor [234], forwhich S can be smaller or even negative [235]. Other situations in which S < 0, suchas loops involving scalars or Majorana particles, are also possible [236]. The simplestorigin of S < 0 would probably be an additional heavy Z ′ boson [223], which couldmimic S < 0. Supersymmetric extensions of the SM generally give very small effects. SeeRefs. 181 and 237 and the note on “Supersymmetry” in the Searches Particle Listings fora complete set of references.

Most simple types of new physics yield U = 0, although there are counter-examples,such as the effects of anomalous triple gauge vertices [228].

The SM expressions for observables are replaced by

M2Z = M2

Z01 − α(MZ)T

1 − GF M2Z0S/2

√2π

,

M2W = M2

W01

1 − GF M2W0(S + U)/2

√2π

, (10.59)

where MZ0 and MW0 are the SM expressions (as functions of mt and MH) in the MS

scheme. Furthermore,

ΓZ =M3

ZβZ

1 − α(MZ)T, ΓW = M3

W βW , Ai =Ai0

1 − α(MZ)T, (10.60)

where βZ and βW are the SM expressions for the reduced widths ΓZ0/M3Z0 and

ΓW0/M3W0, MZ and MW are the physical masses, and Ai (Ai0) is a neutral-current

amplitude (in the SM).The data allow a simultaneous determination of s 2

Z (from the Z pole asymmetries), S(from MZ), U (from MW ), T (mainly from ΓZ), αs (from R�, σhad, and ττ ), and mt

(from CDF and DØ), with little correlation among the SM parameters:

S = 0.01 ± 0.10 (−0.08),

T = 0.03 ± 0.11 (+0.09),

U = 0.06 ± 0.10 (+0.01), (10.61)

July 30, 2010 14:36


and s 2Z = 0.23124 ± 0.00016, αs(MZ) = 0.1183 ± 0.0016, mt = 173.0 ± 1.3 GeV, where

the uncertainties are from the inputs. The central values assume MH = 117 GeV andin parentheses we show the difference to assuming MH = 300 GeV instead. As can beseen, the SM parameters (U) can be determined with no (little) MH dependence. On theother hand, S, T , and MH cannot be obtained simultaneously, because the Higgs bosonloops themselves are resembled approximately by oblique effects. Eqs. (10.61) show thatnegative (positive) contributions to the S (T ) parameter can weaken or entirely removethe strong constraints on MH from the SM fits. Specific models in which a large MH iscompensated by new physics are reviewed in Ref. 238. The parameters in Eqs. (10.61),which by definition are due to new physics only, are in reasonable agreement with theSM values of zero. Fixing U = 0 (as is also done in Fig. 10.4) moves S and T slightlyupwards,

S = 0.03 ± 0.09 (−0.07),

T = 0.07 ± 0.08 (+0.09). (10.62)

The correlation between S and T in this fit amounts to 87%.Using Eq. (10.57), the value of ρ0 corresponding to T in Eq. (10.61) is 1.0002 ±

0.0009 (+0.0007), while the one corresponding to Eq. (10.62) is 1.0006± 0.0006 (+0.0007).The values of the ε parameters defined in Eq. (10.56) are

ε3 = 0.0000± 0.0009 (−0.0006),

ε1 = 0.0002± 0.0008 (+0.0007),

ε2 = −0.0005 ± 0.0009 (−0.0001). (10.63)

Unlike the original definition, we defined the quantities in Eqs. (10.63) to vanishidentically in the absence of new physics and to correspond directly to the parametersS, T , and U in Eqs. (10.61). There is a strong correlation (88%) between the S and Tparameters. The allowed regions in S–T are shown in Fig. 10.4. From Eqs. (10.61) oneobtains S ≤ 0.16 (0.08) and T ≤ 0.21 (0.29) at 95% CL for MH = 117 GeV (300 GeV). Ifone fixes MH = 600 GeV and requires the constraint S ≥ 0 (as is appropriate in QCD-likeTechnicolor models) then S ≤ 0.13 (Bayesian) or S ≤ 0.10 (frequentist). This rules outsimple Technicolor models with many techni-doublets and QCD-like dynamics.

An extra generation of SM fermions is excluded at the 6 σ level on the basis of theS parameter alone, corresponding to NF = 2.85 ± 0.20 for the number of families. Thisresult assumes that there are no new contributions to T or U and therefore that any newfamilies are degenerate, and is in agreement with a fit to the number of light neutrinos,Nν = 2.991± 0.007. However, the S parameter fits are valid even for a very heavy fourthfamily neutrino. This restriction can be relaxed by allowing T to vary as well, since T > 0is expected from a non-degenerate extra family. Fixing S = 2/3π, the global fit favors afourth family contribution to T of 0.22± 0.04. However, the quality of the fit deteriorates(Δχ2 = 3.2 relative to the SM fit with MH forced not to drop below its LEP 2 bound of114.4 GeV) so that this tuned T scenario is also disfavored but only at about the 90%

July 30, 2010 14:36


-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

S

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

T

all: MH = 117 GeV

all: MH = 340 GeV

all: MH = 1000 GeV

ΓZ, σ

had, R

l, R

q

asymmetries

MW

ν scattering

e scattering

APV

Figure 10.4: 1 σ constraints (39.35%) on S and T from various inputs combinedwith MZ . S and T represent the contributions of new physics only. (Uncertaintiesfrom mt are included in the errors.) The contours assume MH = 117 GeV exceptfor the central and upper 90% CL contours allowed by all data, which are forMH = 340 GeV and 1000 GeV, respectively. Data sets not involving MW areinsensitive to U . Due to higher order effects, however, U = 0 has to be assumed inall fits. αs is constrained using the τ lifetime as additional input in all fits. Becausethis has changed significantly since the 2008 edition of this Review (see the discussionin Sec. 10.5), the strongly αs-dependent solid (green) contour from Z lineshapeand cross-section measurements has moved significantly towards negative S and T .The long-dashed (magenta) contour from ν scattering has moved closer towardsthe global averages (see Sec. 10.3). The long-dash-dotted (indigo) contour frompolarized e scattering [123,125] is the upper tip of an elongated ellipse centered ataround S = −15 and T = −21. At first sight it looks as if it is deviating stronglybut it is off by only 1.8 σ. This illusion arises because Δχ2 > 0.8 everywhere on thevisible part of the contour. Color version at end of book.

CL which is weaker compared to the 2008 edition of this Review, i.e. before the latestdevelopments in APV and the ν-DIS interpretation. In fact, tuned mass splittings of theextra leptons and quarks [239] can now yield fits with only moderately higher χ2 values(by about 1 unit) than for the SM. A more detailed analysis is also required if the extraneutrino (or the extra down-type quark) is close to its direct mass limit [240]. Thus, afourth family is disfavored but not excluded by current data. Similar remarks apply to aheavy mirror family [241] involving right-handed SU(2) doublets and left-handed singlets.A more detailed discussion based on the same data set as used for this Review can be

July 30, 2010 14:36


found in Ref. 242. Additional heavy ordinary or mirror generations may also requirelarge Yukawa and Higgs couplings that may lead to Landau poles at low scales [243].In contrast, heavy degenerate non-chiral (also known as vector-like or exotic) multiplets,which are predicted in many grand unified theories [244] and other extensions of the SM,do not contribute to S, T , and U (or to ρ0), and do not require large coupling constants.Such exotic multiplets may occur in partial families, as in E6 models, or as completevector-like families [245].

There is no simple parametrization to describe the effects of every type of newphysics on every possible observable. The S, T , and U formalism describes many typesof heavy physics which affect only the gauge self-energies, and it can be applied to allprecision observables. However, new physics which couples directly to ordinary fermions,such as heavy Z ′ bosons [223], mixing with exotic fermions [246], or leptoquarkexchange [160,247] cannot be fully parametrized in the S, T , and U framework. It isconvenient to treat these types of new physics by parameterizations that are specializedto that particular class of theories (e.g., extra Z ′ bosons), or to consider specific models(which might contain, e.g., Z ′ bosons and exotic fermions with correlated parameters).Fits to Supersymmetric models are described in Refs. 181 and 248. Models involvingstrong dynamics (such as (extended) Technicolor) for electroweak breaking are consideredin Ref. 249. The effects of compactified extra spatial dimensions at the TeV scale arereviewed in Ref. 250, and constraints on Little Higgs models in Ref. 251. The implicationsof non-standard Higgs sectors, e.g., involving Higgs singlets or triplets, are discussed inRef. 252. Limits on new four-Fermi operators and on leptoquarks using LEP 2 and lowerenergy data are given in Refs. [160,253]. Constraints on various types of new physics arereviewed in Refs. [59,128,254,255], and implications for the LHC in Ref. 256.

An alternate formalism [257] defines parameters, ε1, ε2, ε3, and εb in terms of thespecific observables MW /MZ , Γ��, A

(0,�)FB , and Rb. The definitions coincide with those

for εi in Eqs. (10.55) and (10.56) for physics which affects gauge self-energies only, butthe ε’s now parametrize arbitrary types of new physics. However, the ε’s are not relatedto other observables unless additional model-dependent assumptions are made. Anotherapproach [258] parametrizes new physics in terms of gauge-invariant sets of operators.It is especially powerful in studying the effects of new physics on non-Abelian gaugevertices. The most general approach introduces deviation vectors [254]. Each type ofnew physics defines a deviation vector, the components of which are the deviations ofeach observable from its SM prediction, normalized to the experimental uncertainty. Thelength (direction) of the vector represents the strength (type) of new physics.

One of the best motivated kinds of physics beyond the SM besides Supersymmetryare extra Z ′ bosons [259]. They do not spoil the observed approximate gauge couplingunification, and appear copiously in many Grand Unified Theories (GUTs), mostSuperstring models [260], as well as in dynamical symmetry breaking [249] and LittleHiggs models [251]. For example, the SO(10) GUT contains an extra U(1) as canbe seen from its maximal subgroup, SU(5) × U(1)χ. Similarly, the E6 GUT containsthe subgroup SO(10) × U(1)ψ. The Zψ possesses only axial-vector couplings to theordinary fermions, and its mass is generally less constrained. The Zη boson is the linear

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Table 10.10: 95% CL lower mass limits (in GeV) from low energy and Z poledata on various extra Z ′ gauge bosons, appearing in models of unification andstring theory. More general parametrizations are described in Refs. 259,264. Theelectroweak results [265] are for Higgs sectors consisting of doublets and singletsonly (ρ0 = 1) with unspecified U(1)′ charges. The CDF [266] and DØ [267] boundsfrom searches for pp → μ+μ− and e+e−, respectively, are listed in the next twocolumns, followed by the LEP 2 e+e− → f f bounds [160] (assuming θ = 0). (TheTevatron bounds would be moderately weakened if there are open supersymmetricor exotic decay channels [268]) . The last column shows the 1 σ ranges for MHwhen it is left unconstrained in the electroweak fits.

Z ′ electroweak CDF DØ LEP 2 MH

Zχ 1, 141 892 800 673 171+493−89

Zψ 147 878 763 481 97+31−25

Zη 427 982 810 434 423+577−350

ZLR 998 630 − 804 804+174−35

ZS 1, 257 821 719 − 149+353−68

ZSM 1, 403 1, 030 950 1, 787 331+669−246

Zstring 1, 362 − − − 134+209−58

combination√

3/8Zχ −√5/8Zψ. The ZLR boson occurs in left-right models with gauge

group SU(3)C × SU(2)L × SU(2)R × U(1)B−L ⊂ SO(10), and the secluded ZS emergesin a supersymmetric bottom-up scenario [261]. The sequential ZSM boson is defined tohave the same couplings to fermions as the SM Z boson. Such a boson is not expectedin the context of gauge theories unless it has different couplings to exotic fermions thanthe ordinary Z boson. However, it serves as a useful reference case when comparingconstraints from various sources. It could also play the role of an excited state of theordinary Z boson in models with extra dimensions at the weak scale [250]. Finally,we consider a Superstring motivated Zstring boson appearing in a specific model [262].The potential Z ′ boson is in general a superposition of the SM Z and the new bosonassociated with the extra U(1). The mixing angle θ satisfies,

tan2 θ =M2

Z01− M2

Z

M2Z′ − M2

Z01

,

where MZ01

is the SM value for MZ in the absence of mixing. Note, that MZ < MZ01,

and that the SM Z couplings are changed by the mixing. The couplings of the heavier Z ′may also be modified by kinetic mixing [259,263]. If the Higgs U(1)′ quantum numbers

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are known, there will be an extra constraint,

θ = Cg2

g1

M2Z

M2Z′

,

where g1,2 are the U(1) and U(1)′ gauge couplings with g2 =√

53 sin θW

√λ g1 and

g1 =√

g2 + g′2. λ ∼ 1 (which we assume) if the GUT group breaks directly toSU(3)×SU(2)×U(1)×U(1)′. C is a function of vacuum expectation values. For minimalHiggs sectors it can be found in Ref. 223. Table 10.10 shows the 95% CL lower masslimits [265] for ρ0 = 1 and 114.4 GeV ≤ MH ≤ 1 TeV. The last column shows the 1 σranges for MH when it is left unconstrained. In cases of specific minimal Higgs sectorswhere C is known, the Z ′ mass limits are generally pushed into the TeV region. The limitson |θ| are typically smaller than a few ×10−3. For more details see [259,265,269] and thenote on “The Z ′ Searches” in the Gauge & Higgs Boson Particle Listings. Also listed inTable 10.10 are the direct lower limits on Z ′ production from the Tevatron [266,267] andLEP 2 bounds [160].

Acknowledgments:

This work was supported in part by CONACyT (Mexico) contract 82291–F by an IBMEinstein Fellowship, and by NSF grant PHY–0503584.

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